ARS MATHEMATICA CONTEMPORANEA Volume 12, Number 1, Spring/Summer 2017, Pages 1-203 Covered by: Mathematical Reviews Zentralblatt MATH COBISS SCOPUS Science Citation Index-Expanded (SCIE) Web of Science ISI Alerting Service Current Contents/Physical, Chemical & Earth Sciences (CC/PC & ES) The University of Primorska The Society of Mathematicians, Physicists and Astronomers of Slovenia The Institute of Mathematics, Physics and Mechanics The publication is partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications. ARS MATHEMATICA CONTEMPORANEA Slovenian Discrete and Applied Mathematics Society We are very pleased to announce the formation of the Slovenian Discrete and Applied Mathematics Society, in Koper (Slovenia), on 14 December 2016. The aim of this society is to promote the mathematical sciences, with special emphasis given to discrete and applied mathematics. 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Dragan Marušic and Tomaž Pisanski Editors In Chief iii ARS MATHEMATICA CONTEMPORANEA Contents A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs Simona Bonvicini, Tomaž Pisanski...................... 1 The parameters of Fibonacci and Lucas cubes Aleksandar Ilic, Marko Milosevic ......................25 A characterization of plane Gauss paragraphs Dan Archdeacon, Drago Bokal, Tanja Gologranc.............. 31 On skew Heyting algebras Karin Cvetko-Vah............................... 37 A decomposition for Markov processes at an independent exponential time Mihael Perman................................ 51 Euler's divergent series and an elementary model in Statistical Physics Bill Allombert, Jean-Paul Simon Allouche, Michel Mendes France.....67 The distinguishing index of the Cartesian product of finite graphs Aleksandra Gorzkowska, Rafal Kalinowski, Monika Pilsniak........77 Xd (G), |Aut(G)|, and a variant of the Motion Lemma Niranjan Balachandran, Sajith Padinhatteeri................. 89 Coherent configurations over copies of association schemes of prime order Reza Sharafdini, Mitsugu Hirasaka......................111 The strong metric dimension of generalized Sierpiiiski graphs with pendant vertices Ehsan Estaji, Juan Alberto Rodriguez-Velazquez ..............127 Remarks on the thickness of Kn,n,n Yan Yang..........' .'........................135 Automorphism group of the balanced hypercube Jin-Xin Zhou, Jin Ho Kwak, Yan-Quan Feng, Zhen-Lin Wu.........145 Uniquely colorable Cayley graphs Walter Klotz, Torsten Sander.........................155 Non-negative spectrum of a digraph Irena M. Jovanovics .............................. 167 Cycle bases of reduced powers of graphs Richard H. Hammack, Gregory D. Smith...................183 Volume 12, Number 1, Spring/Summer 2017, Pages 1-203 v ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 1-24 A novel characterization of cubic Hamiltonian graphs via the associated quartic graphs* Simona Bonvicini t Department of Sciences and Methods for Engineering, University ofModena and Reggio Emilia, Italy Tomaž Pisanski University of Primorska, Koper and University of Ljubljana, Ljubljana, Slovenia Received 16 August 2015, accepted 18 December 2015, published online 6 March 2016 We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is most useful in the case when it induces a blue and red 2-factorization of the associated quartic graph. We use this condition to characterize the Hamiltonian I-graphs, a further generalization of generalized Petersen graphs. The characterization of Hamiltonian I-graphs follows from the fact that one can choose a 1-factor in any I-graph in such a way that the corresponding associated quartic graph is a graph bundle having a cycle graph as base graph and a fiber and the fundamental factorization of graph bundles playing the role of blue and red factorization. The techniques that we develop allow us to represent Cayley multigraphs of degree 4, that are associated to abelian groups, as graph bundles. Moreover, we can find a family of connected cubic (multi)graphs that contains the family of connected I-graphs as a subfamily. Keywords: Generalized Petersen graphs, I-graphs, Hamiltonian cycles, Eulerian tours, Cayley multi-graphs. Math. Subj. Class.: 05C45, 05C25, 05C15, 05C76, 05C70, 55R10, 05C60 * Research supported in part by the ARRS Grants P1-0294, N1-0032, and J1-6720. t Corresponding author E-mail addresses: simona.bonvicini@unimore.it (Simona Bonvicini), Tomaz.Pisanski@fmf.uni-lj.si (Tomaz Abstract Pisanski) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Ars Math. Contemp. 12 (2017) 25-29 1 Introduction A graph is Hamiltonian if it contains a spanning cycle (Hamiltonian cycle). To find a Hamiltonian cycle in a graph is an NP-complete problem (see [12]). This fact implies that a characterization result for Hamiltonian graphs is hard to find. For this reason, most graph theorists have restricted their attention to particular classes of graphs. In this paper we consider cubic graphs. In Section 2 we give a necessary and sufficient condition for a cubic graph to be Hamiltonian. Using this condition we can completely characterize the Hamiltonian I-graphs. The family of I-graphs is a generalization of the family of generalized Petersen graphs. In [5], the generalized Petersen graphs were further generalized to I-graphs. Let n, p, q be positive integers, with n > 3, 1 < p, q < n - 1 and p, q = n/2. An I-graph I(n,p, q) has vertex-set V(I(n,p, q)) = {v^u : 0 < i < n - 1} and edge-set E(1 (n,p, q)) = {[vj, vj+p], [vj, Wj], [w, ui+q] : 0 < i < n - 1} (subscripts are read modulo n). The graph I(n,p, q) is isomorphic to the graphs I(n, q,p), I(n, n - p, q) and I(n,p, n - q). It is connected if and only if gcd(n,p, q) = 1 (see [3]). For p =1 the I-graph I(n, 1, q) is known as a generalized Petersen graph and is denoted by G(n, q). The Petersen graph is G(5,2). It has been proved that I(n,p, q) is isomorphic to a generalized Petersen graph if and only if gcd(n,p) = 1 or gcd(n, q) = 1 (see [3]). A connected I-graph which is not a generalized Petersen graph is called a proper I-graph. Recently, the class of I-graphs has been generalized to the class of GI-graphs (see [6]). It is well known that the Petersen graph is not Hamiltonian. A characterization of Hamiltonian generalized Petersen graphs was obtained by Alspach [2]. Theorem 1.1 (Alspach, [2]). A generalized Petersen graph G(n, q) is Hamiltonian if and only if it is not isomorphic to G(n, 2) when n = 5 (mod 6). In this paper we develop a powerful theory that helps us to extend this result to all I-graphs. Theorem 1.2. A connected I-graph is Hamiltonian if and only if it is not isomorphic to G(n, 2) when n = 5 (mod 6). For the proof of the above main theorem, we developed techniques that are of interest by themselves and are presented in the following sections. In particular, we introduce good Eulerian graphs that are similar to lattice diagrams that were originally used by Alspach in his proof of Theorem 1.1. Our theory also involves Cayley multigraphs. In Section 4 we show that Cayley multi-graphs of degree 4, that are associated to abelian groups, can be represented as graph bundles [19]. By the results concerning the isomorphisms between Cayley multigraphs (see [7]), we can establish when two graph bundles are isomorphic or not (see Section 4.2). Combining the definition of graph bundles with Theorem 3.3, we can find a family of connected cubic (multi)graphs that contains the family of connected I-graphs as a subfamily (see Section 5). S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 3 2 Cubic graph with a 1-factor and the associated quartic graph with transitions A cubic Hamiltonian graph has a 1-factor. In fact, it has at least three (edge-disjoint) 1-factors. Namely any Hamiltonian cycle is even and thus gives rise to two 1-factors and the remaining chords constitute the third 1-factor. The converse is not true. There are cubic graphs, like the Petersen graph, that have a 1-factor but are not Hamiltonian. Nevertheless, we may restrict our search for Hamiltonian graphs among the cubic graphs to the ones that possess a 1-factor. In this section, we give a necessary and sufficient condition for the existence of a Hamiltonian cycle in a cubic graph G possessing a 1-factor F. Let G be a connected simple cubic graph and let F be one of its 1-factors. Denote by X = G/F the graph obtained from G by contracting the edges of F. The graph X is connected, quartic, that is, regular of degree 4 and might have multiple edges (X has no loop since G is simple). We say that the quartic graph X is associated with G and F. Since X is even and connected, it is Eulerian. A path on three vertices with middle vertex v that is a subgraph of X is called a transition at v. Since any pair of edges incident at v defines a transition, there are (2) = 6 transitions at each vertex of X. For general graphs each vertex of valence d gives rise to (2) transitions. In an Euler tour some transitions may be used, others are not used. We are interested in some particular Eulerian spanning subgraphs W. Note that any such graph is sub-quartic and the valence at any vertex of W is either 4 or 2. A vertex of valence 4 has therefore 6 transitions, while each vertex of valence 2 has (2) = 1 transition. Let Y be the complementary 2-factor of F in G. Note that the edges of Y are in one-to-one correspondence with the edges of X, while the edges of F are in one-to-one correspondence with the vertices of X. If a is an edge of Y, we denote by a' the corresponding edge in X. If e is an edge of F, the corresponding vertex of X will be denoted by xe. Let u and v be the end-vertices of e and let a and b be the other edges incident with u and similarly c and d the edges incident with v. After contraction of e, the vertex xe is incident with four edges: a', b', c', d'. By considering the pre-images of the six transitions at xe, they fall into two disjoint classes. Transitions a'b' and c'd' are non-traversing while the other four transitions are traversing transitions. In the latter case the edge e has to be used to traverse from one edge of the pre-image transition to the other. Let W be a spanning Eulerian subgraph of X. Transitions of X carry over to W. The 4-valent vertices of W keep the same six transitions, while each 2-valent vertex inherits a single transition. We say that W is admissible if the transition at each 2-valent vertex of W is traversing. Let W be an admissible Eulerian subgraph of X. A closed walk in W that allows only non-traversing transitions at each 4-valent vertex of W is said to be a closed walk with allowed transitions. A closed walk with allowed transitions passing through a 4-valent vertex xe of W might use both transitions a'b', c'd' or only one of the two non-traversing transitions. If it passes through a 2-valent vertex of W, then it uses traversing transitions. Hence, the underlying graph of a closed walk with allowed transitions might be a cycle. A partition of the edge-set of W into closed walks with allowed transitions is said to be a tour with allowed transitions. Each closed walk in the tour is a component of the tour. Lemma 2.1. Let G be a connected cubic graph with 1-factor F. There is a one-to-one correspondence between 2-factors T of G and admissible Eulerian subgraphs W of X = G/F in such a way that the number of cycles of T is the same as the number of components of a tour with allowed transitions in W. 4 Ars Math. Contemp. 12 (2017) 25-29 Proof. Let T be a 2-factor of G and let e = uv be an edge of the 1-factor F. Let W be the projection of T to X = G/F. We will use the notation introduced above. Hence the edge e and its end-vertices u and v project to the same vertex xe of X. There are two cases: Case 1: e belongs to T. In this case exactly one other edge, say a, incident with u and another edge, say c, incident with v belong to T. The other two edges (b and d) do not belong to T. This means that xe is a 2-valent vertex with traversing transition. Case 2: e does not belong to T. In this case both edges a and b incident with u belong to T and both edges c and d incident with v belong to T. In this case xe is a 4-valent vertex with non-traversing transitions. Clearly, W is an admissible Eulerian subgraph. Each component of the tour determined by W with transitions gives back a cycle of T. The correspondence between T and W is therefore established. □ An Eulerian tour in W with allowed transitions is said to be good. An admissible subgraph W of X possessing a good Eulerian tour is said to be a good Eulerian subgraph. In a good Eulerian subgraph W there are two extreme cases: 1. each vertex of W is 4-valent: this means that W = X; in this case the complementary 2-factor Y = G - F is a Hamiltonian cycle and no edge of F is used; 2. each vertex of W is 2-valent: this means that W is a good Hamiltonian cycle in X. In this case F together with the pre-images of edges of W in G form a Hamiltonian cycle. Theorem 2.2. Let G be a connected cubic graph with 1-factor F. Then G is Hamiltonian if and only if X = G/F contains a good Eulerian subgraph W. Proof. Clearly G is Hamiltonian if and only if it contains a 2-factor with a single cycle. By Lemma 2.1, this is true if and only if W is an admissible Eulerian subgraph possessing an Eulerian tour with allowed transitions. But this means W is good. □ Corollary 2.3. Let G be a connected cubic graph with 1-factor F. Finding a good Eulerian subgraph W of X = G/F is NP-complete. Proof. Since finding a good Eulerian subgraph is equivalent to finding a Hamiltonian cycle in a cubic graph, and the latter is NP-complete [12], the result follows readily. □ Also in [11] Eulerian graphs are used to find a Hamiltonian cycle (and other graph properties), but our method is different. The results of this section may be applied to connected I-graphs. The obvious 1-factor F of an I-graph I(n,p, q) consists of spokes. Let Q(n,p, q) denote the quotient I(n,p, q)/F. We will call Q(n,p, q) the quartic graph associated with I(n,p, q). Corollary 2.4. Let I(n,p, q) be a connected I-graph and let Q(n,p, q) be its associated quartic graph. Then I(n,p, q) is Hamiltonian if and only if Q(n,p, q) contains a good Eulerian subgraph W. S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 5 3 Special 1-factors and their applications Let G be a cubic graph, F a 1-factor and Y the complementary 2-factor of F in G. Define an auxiliary graph Y(G, F) having cycles of Y as vertices and having two vertices adjacent if and only if the corresponding cycles of Y are joined by one or more edges of F. If an edge of F is a chord in one of the cycles of Y, then the graph Y(G, F) has a loop. We shall say that the 1-factor F is special if the graph Y(G, F) is bipartite. A cubic graph with a special 1-factor will be called special. If F is a special 1-factor of G, then the edges of F join vertices belonging to distinct cycles of Y since Y(G, F) is loopless. Theorem 3.1. Let G be a connected cubic graph with a 1-factor, and let F be one of its 1-factors and X = G/F its associated quartic graph. Then X admits a 2-factor whose edges may be colored blue or red in such a way that the traversing transitions are exactly color-switching and non-traversing transitions are color-preserving if and only if G and F are special. Proof. Assume that F is a special 1-factor of G. Since Y(G, F) is bipartite, we can bicolor the vertices of Y(G, F): let one set of the bipartition be blue and the other red. This coloring induces a coloring on the edges of Y: for every blue vertex (respectively, red vertex) of Y(G, F) we color in blue (respectively, in red) the edges of the corresponding cycle of Y. Since the edges of Y are in one-to-one correspondence with the edges of X, we obtain a 2-factorization of X into a blue 2-factor and red 2-factor. Since F is special, the edges of F are incident with vertices of G belonging to cycles of Y with different colors (a blue cycle and a red cycle). Therefore, a traversing transition is color-switching and a non-traversing transition is color-preserving. Conversely, assume that X has a blue and red 2-factorization such that the traversing transitions are color-switching and non-traversing transitions are color-preserving. Since the edges of X are in one-to-one correspondence with the edges of Y, we can partition the cycles of Y into red cycles and blue cycles. Since the traversing transitions are color-switching and non-traversing transitions are color-preserving, the edges of F are incident with edges belonging to cycles of different colors. This means that the graph Y(G, F) is bipartite, hence F is special. □ Proposition 3.2. Let G and F be special and let W be any Eulerian subgraph of X = G/F the associated quartic graph with a blue and red 2-factorization. Then W is admissible if and only if each 2-valent vertex is incident with edges of different colors. Proof. An Eulerian subgraph W is admissible if and only if each 2-valent vertex v in W is incident with edges forming a traversing transition at v. By Theorem 3.1, a traversing transition is color-switching. Hence, W is admissible if and only if the edges incident with v have different colors. □ Note that quartic graphs with a given 2-factorization can be put into one-to-one correspondence with special cubic graphs. Theorem 3.3. Any special cubic graph G with a special 1-factor F gives rise to the associated quartic graph with a blue and red 2-factorization. However, any quartic graph with a given 2-factorization determines back a unique special cubic graph by color-preserving splitting vertices and inserting a special 1-factor. 6 Ars Math. Contemp. 12 (2017) 25-29 Proof. By Theorem 3.1, a special cubic graph G with a special 1-factor F gives rise to the graph X = G/F admitting a blue and red 2-factorization. Conversely, it is well known that every quartic graph X possesses a 2-factorization, that is, the edges of X can be partitioned into a blue and red 2-factor. We use the blue and red 2-factors of X to construct a cubic graph G as follows: put in G a copy of the blue 2-factor and a copy of the red 2-factor; construct a 1-factor of G by joining vertices belonging to distinct copies. It is straightforward to see that G and F are special. □ We will now apply this theory to the I-graphs. In Section 7 we will see that this theory allows us to find a Hamiltonian cycle in a proper I-graph and also to find a family of special cubic graphs that contains the family of I-graphs as a subfamily (see Section 5). Let I(n,p, q) be an I-graph. A vertex vi (respectively, u) is called an outer vertex (respectively, an inner vertex). An edge of type [v, vi+p] (respectively, of type [u, ui+q]) is called an outer edge (respectively, an inner edge). An edge [vi, ui] is called a spoke. The spokes of I(n,p, q) determine a 1-factor of I(n,p, q). The set of outer edges is called the outer rim, the set of inner edges is called the inner rim. As a consequence of the results proved in [3], the following holds. Proposition 3.4. Let I(n,p, q), n > 3, 1 < p, q < n — 1, p, q = n/2, be an I-graph. Set t = gcd(n, q) and s = n/t. Then t < n/2 and 3 < s < n. Moreover, I(n,p, q) is connected if an only if gcd(t,p) = 1 and gcd(s,p) is coprime with q. It is proper if and only if t and gcd(s,p) are different from 1. Proof. The integer t satisfies the inequality t < n/2, since t is a divisor of q and q < n — 1, q = n/2; whence 3 < s < n. By the results proved in [3], I(n,p, q) is connected if and only if gcd(n,p, q) = 1. Since n = st and q = t(q/t), the relation gcd(n,p, q) = 1 can be written as gcd(st,p, t(q/t)) = 1, whence gcd(t,p) = 1 and gcd(s,p) is coprime with q. Also the converse is true, and therefore I(n, p, q) is connected if and only if gcd(t, p) = 1 and gcd(s,p) is coprime with q. A connected I-graph I(n,p, q) is a generalized Petersen graph if and only if gcd(n, q) = 1 or gcd(n,p) = 1 (see [3]). By the previous results, I(n,p, q) is a generalized Petersen graph if and only if 1 = gcd(n, q) = t or 1 = gcd(n,p)= gcd(st,p)= gcd(s,p). The assertion follows. □ The smallest proper I-graphs are I(12, 2, 3) and I(12,4,3). It is straightforward to see that the following result holds. Lemma 3.5. Let F be the 1-factor determined by the spokes of I(n,p, q) and X = Q(n,p, q) its associated quartic graph. Then F is special, the graph X is a circulant multigraph Cir (n; p, q), the blue edges of X correspond to the inner rim and the red edges to the outer rim of I(n,p, q). In the next section we introduce a class of graphs X(s,t, r) and later show that it contains Q(n,p, q) as its subclass. 4 Graphs X (s,t,r) Let r be a group in additive notation with identity element 0r. Let S be a list of not necessarily distinct elements of r satisfying the symmetry property S = —S = {—7 : 7 G S}. The Cayley multigraph associated with r and S, denoted by Cay(r, S), is an undirected multigraph having the elements of r as vertices and edges of the form [x, x + 7] with S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 7 x e r, y e S .If r is a cyclic group of order n, then Cay (r, S) is a circulant multigraph of order n. A Cayley multigraph Cay(r, S) is regular of degree |S| (in determining |S|, each element of S is considered according to its multiplicity in S). It is connected if and only if S is a set of generators of r. If the elements of S are pairwise distinct, then Cay(r, S) is a simple graph and we will use the term Cayley graph. We are interested in connected Cayley multigraphs of degree 4. In this case we write S as the list S = {±yi, ±Y2}. A circulant multigraph of order n will be denoted by Cir(n; ±yi , ±y2). If Y®, with i e {1,2}, is an involution of r or the trivial element, then ±y® means that the element y® appears twice in the list S. Consequently, the associated Cayley multigraph has multiple edges or loops. We will denote by o(y®) the order of y®. We will show that the Cayley multigraphs Cay(r, {±yi, ±Y2}) defined on a suitable abelian group r (and in particular the circulant multigraphs Cir(n; ±yi, ±y2)) can be given a different interpretation in terms of X(s, t, r) graphs (see Figure 1) defined as follows. Definition 4.1. Let s, t > 1, 0 < r < s — 1 be integers. Let X(s, t, r) be the graph with vertex-set {xj :0 < i < t —1,0 < j < s — 1} and edge-set {[xj ,xj+1] :0 < i < t —1,0 < j < s — 1}U {[xj,xj+1] : 0 < i < t — 2,0 < j < s — 1}U {[xj-1,x0+r] : 0 < j < s — 1} (the superscripts are read modulo t, the subscripts are read modulo s). The graph X (s, t, r) has edges of type [xj, xj+1], [xj, xj+1] or [xj-1, x0+r]. An edge of type [xj, xj+1] will be called horizontal. An edge of type [xj, xj+1] will be called vertical, an edge of type [xj-1, x0+r ] will be called diagonal. For t = 1, we say that the edges are horizontal and diagonal (a diagonal edge is an edge of type [x0, x0+r]). For s = 1, the horizontal edges are loops. For (t, r) = (1,0), the diagonal edges are loops. For s = 2 or s > 2 and (t, r)=(1,1), (1, s/2), (2,0) the graph has multiple edges. For the other values of s, t, r, the graph X(s, t, r) is a simple graph. A simple graph X(s, t, r) is a graph bundle with a cycle fiber Cs over a cycle base Ct; the parameter r represents an automorphism of the cycle Cs that shifts the cycle r steps (see [19] for more details on graph bundles). In the literature a simple graph X(s, t, r) is also called r-pseudo-cartesian product of two cycles (see for instance [10]). The definition of X(s,t, r) suggests that the graph X(s,t, r) is isomorphic to X(s, t, s — r). The existence of this isomorphism can be also obtained from the following statement. Figure 1: The graph X(s,t, r) is embedded into torus with quadrilateral faces; it has a blue and red 2-factorization: the vertical and diagonal edges form the blue 2-factor, the horizontal edges form the red 2-factor. 8 Ars Math. Contemp. 12 (2017) 25-29 Proposition 4.2. Let Cay(r, {±71, ±72}) be a connected Cayley multigraph of degree 4, where r is an abelian group, o(y1) = s and |r|/s = t. Then aY2 = rY1 for some integer r, 0 < r < s — 1, if and only if a = t. Consequently, Cay(r, {±y1, ±y2}) can be represented as the graph X(s, t, r) or X(s, t, s — r). Proof. We show that G1 = Cay(r, {±y1, ±y2}) and G2 = X(s,t,r) are isomorphic. Since y1 and y2 are generators of r, the elements of r can be written in the form iY2 + jY1, where iY2 € (y2), jY € (y1). Hence we can describe the elements of r by the left cosets of the subgroup (y1) in r. By this representation, we can see that the endvertices of an edge [x, x ± y1] of Cay(r, {±y1, ±Y2}) belong to the same left coset of (y1) in r; the endvertices of an edge [x, x ± y2] belong to distinct left cosets of (y1) in r. Therefore, aY2 = rY1 € (y1) if and only if a = t, since Cay(r, {±y1, ±Y2}) is connected and |F/(y1)| = t. Hence we can set V(G1) = {¿Y2 + jY1 : 0 < i < t — 1,0 < j < s — 1} and E(G1) = {[iY2 + jY1, (i + 1)Y2 + jY1], [iY2 + jY1, iY2 + (j + 1)Y1] : 0 < i < t — 1, 0 < j < s — 1}. The map ^ : V(G1) ^ V(G2) defined by ^(iY2 + jY1) = xj is a bijection between V(G1) and V(G2). Moreover, if v1, v2 are adjacent vertices of G1, that is, v1 = iY2 + jY1 and V2 = (i + 1)y2 + jY1 (or «2 = iY2 + (j + 1)Y1), then ^(«1) = xj, ^(«2) = xj+1 (or y(v2) = xj+1) are adjacent vertices of G2. In particular, if v1 = (t — 1)y2 + jY1 and «2 = tY2 + jY1= rY1 + jY1= (r + j)Y1, then ^(«1) = xj-1, ^(«2) = x0+j are adjacent vertices of G2. It is thus proved that ^ is an isomorphism between G1 and G2. If we replace the element y1 by its inverse —y1, then G1 is the graph X(s, t, s — r). □ In what follows, we show that for s, t > 1 there exists a Cayley multigraph on a suitable abelian group that satisfies Proposition 4.2, that is, for every s, t > 1 the graph X(s, t, r) can be represented as a Cayley multigraph. The proof is particularly easy when t = 1; r = 0; or s = 2. For these cases, the following holds. Proposition 4.3. The graph X(s, 1,r), with s > 1, 0 < r < s — 1, is the circulant multigraph Cir(s; ±1, ±r). The graph X(s, t, 0), with s, t > 1, is the Cayley multigraph Cay(Zs x Zt, {±(1,0), ±(0,1)}). The graph X(2, t, 1), with t > 1, is the circulant multigraph Cir(2t; ±t, ±1). Proof. For the graph X(s, 1, r) we apply Proposition 4.2 with r = Zst, y1 = 1, Y2 = r. For the graph X(s,t, 0) we apply Proposition 4.2 with r = Zs x Zt, y1 = (1,0), Y2 = (0,1). For the graph X(2,t, 1) we apply Proposition 4.2 with r = Z2t, y1 = t, Y2 = 1. □ The following lemmas concern the graph X (s,t,r) with s > 3, t > 2 and 0 < r < s — 1. They will be used in the proof of Proposition 4.6. Lemma 4.4. Let a > 1 be an integer and let b > 1 be a divisor of a. Let {[c]b : 0 < c < b — 1} be the residue classes modulo b. Every equivalence class [c]b whose representative c is coprime with b contains at least one integer h, 1 < h < a — 1, such that gcd(a, h) = 1. Proof. The assertion is true if b = a (we set h = c). We consider b < a. Let [c]b be an equivalence class modulo b with 1 < c < b — 1 and gcd(c, b) = 1. If c is coprime with a, then we set h = c and the assertion follows. We consider the case gcd(c, a) = 1. We denote by P the set of distinct prime numbers dividing a. We denote by (respectively, by Pc) the subset of P containing the prime numbers dividing b (respectively, c). Since b is a divisor of a (respectively, gcd(c, a) = 1), the set Pb (respectively, Pc) is non-empty. S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 9 Since c and b are coprime, the subsets Vb, Vc are disjoint. We set V' = V \ (Vb U Vc). The set V' might be empty. We denote by w the product of the prime numbers in V' (if V' is empty, then we set w = 1) and consider the integer h = c + wb G [c]b. We show that h < a. Note that w < a/(2b). More specifically, (a/b) > dl^pp) • w > 2w, whence w < a/(2b). Hence h = c + wb < c + (a/2) < a, since c < b and b < (a/2). One can easily verify that gcd(h, a) = gcd(c + wb, a) = 1, since no prime number in V = V' U Vb U Vc can divide c + wb. The assertion follows. □ Lemma 4.5. Let s > 3, t > 2 and Zst/dl be the cyclic group of integers modulo st/d^ where di > 1 is a divisor of d = gcd(s, t). Let (t/di) be the cyclic subgroup of Zst/dl generated by the integer t/di and lei x + (t/di), y + (t/di) be left cosets of (t/di) in ZstM. If x, y G Zsi/dl are congruent modulo d/di, then t(x + (t/di)) = {tx + ^t2/di : 0 < p < s — 1} andt(y + (t/di)) = {ty + ^'t2/di : 0 < p' < s — 1} represent the same subset of Zst/di. Proof. Set x = y + Ad/di with A G Z and t = sm' + m with m' G Z and 0 < m < s — 1. Since gcd(s, t) = d, then also gcd(s, m) = d. Hence the integers dt/di, mt/di G Zst/dl generate the same cyclic subgroup of (t/di) of order s/d. Since t2/di = (sm'+m)t/di = mt/di (mod st/di), each set t(x + (t/di)), t(y + (t/di)) consists of exactly s/d distinct elements of Zst/dl, namely, t(x + (t/di)) = {tx + pmt/dj^ : 1 < p < s/d}, t(y + (t/di)) = {ty + p'mt/dj^ : 1 < p' < s/d}. Therefore, to prove that t(x + (t/di)) = t(y + (t/di)), it suffices to show that every element of t(x + (t/di)) is contained in t(y + (t/di)). Consider tx + pmt/di G t(x + (t/di)). Since x = y + Ad/di, we can write tx + pmt/di = t(y + Ad/di) + pmt/di, whence tx + pmt/di = ty + Adt/di + pmt/di. Since (dt/di) = (mt/di), we can set Adt/di + pmt/di = p'mt/di (mod st/di), with 0 < p' < s/d. Hence tx + pmt/di = ty + p'mt/di (mod st/di), that is, tx + pmt/di G t(y + (t/di)). The assertion follows. □ Proposition 4.6. Let s > 3, t > 2, 0 1, then the equation ax = b (mod n) admits a solution if and only if c is a divisor of b and in this case x = (a/c)-i(b/c) (mod n/c) is a solution to the equation. The following holds. 10 Ars Math. Contemp. 12 (2017) 25-29 Proposition 4.7. Let s, t > 1, 0 < r < s — 1 and d1 = gcd(s, t, r). If r = 0, then there exists an integer k, 0 < k < st/di, such that gcd(k,t) = 1 and k = r/di (mod s/di). The graph X(s,t, r), with r = 0, is isomorphic to the graph X(st/ gcd(s,r), gcd(s, r), r'), where r' = ±t(kdi/gcd(s,r))-1 (mod st/gcd(s, r)). The graph X(s,t, 0) is iso-morphic to the graph X(t, s, 0). Proof. We prove the assertion for s > 3, t > 2 and 0 < r < s — 1. The existence of the integer k follows from Proposition 4.6. By the same proposition, we can represent the graph X(s,t, r) as Cay(Zst/di x Zdl, {±(t/di, 1), ±(k, 0)}). We apply Proposition 4.2 by setting r = Zst/d1 x Zdl, y1 = (k, 0) and y2 = (t/di, 1). Note that gcd(st/di, k) = gcd(s/di, k) = gcd(s, r)/d1, as k is coprime with t and k = r/d1 (mod s/d1). Whence the element (k, 0) has order s' = st/(d1 gcd(st/d1, k)) = st/ gcd(s, r) and t' = |r/((k, 0))| = gcd(s, r). By Proposition 4.2, gcd(s, r)(t/d1,1) = r'(k, 0) for some integer r', 1 < r' < st/ gcd(s, r). The integer r' is a solution to the equation gcd(s, r)(t/d1) = r'k (mod st/d1). By the Chinese Remainder Theorem, r' = t(kd1/ gcd(s, r))-1 (mod st/ gcd(s, r)). An easy calculation shows that s' — r' = —t(kd1/ gcd(s, r))-1 (mod st/ gcd(s, r)). It is straightforward to see that X(s, t, r) and X(s', t', r'), X(s', t', s' — r') are isomorphic. Hence the assertion follows. For the remaining values of s, t, r, we represent the graph X(s, t, r) as the Cayley multigraph in Proposition 4.3 and use Proposition 4.2. Note: if r = 0, then k = r; if r = 0, then set y1 = (0,1), y2 = (1,0) and apply Proposition 4.2. □ 4.1 Fundamental 2-factorization of X(s, t, r) From the definition of X(s,t, r) one can see that the horizontal edges form a 2-factor (the red 2-factor) whose complementary 2-factor in X(s, t, r) is given by the vertical and diagonal edges (the blue 2-factor). We say that the red and blue 2-factor constitute the fundamental 2-factorization of X(s, t, r). A graph X(s, t, r) can be represented as a Cayley multigraph Cay(r, {±y1, ±y2}), where r and {±y1, ±y2} can be defined as in Proposition 4.3 or 4.6. From the proof of the propositions, one can see that the set of horizontal edges of X(s, t, r) is the set {[x, x ± y1 ] : x G r}, the set of vertical and diagonal edges is the set {[x,x ± y2] : x G r}. The edges in {[x,x ± y1] : x G r} will be called the Y1-edges and the edges in the set {[x, x ± y2] : x G r} will be called the Y2-edges. The following result holds. Proposition 4.8. The red 2-factor of X (s, t, r) has exactly t cycles of length s consisting of Y1-edges. The blue 2-factor of X (s, t, r) has exactly gcd(s, r) cycles of length st/ gcd(s, r) consisting of y2 -edges. Proof. It is straightforward to see that the red 2-factor has t horizontal cycles of length s (if s = 1, then each cycle is a loop; if s = 2, then each cycle is a dipole with 2 parallel edges). By the previous remarks, each cycle consists of y1 -edges. The blue 2-factor of X(s, t, r) corresponds to the red 2-factor of the graph X(st/ gcd(s, r), gcd(s, r), r') in Proposition 4.7. Hence it has gcd(s, r) cycles of length st/ gcd(s, r) consisting of Y2-edges. □ 4.2 Isomorphisms between X(s, t, r) graphs We now ask when two graphs X(s,t, r) and X(s',t',r') are isomorphic. Our question is connected to the following well-known problem [7, 14]. Given two isomorphic Cayley S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 11 multigraphs Cay(T, S), Cay(T', S') or, equivalently, given two Cayley representations (r, S), (r', S') of the same multigraph, determine whether (r, S) and (r', S') are equivalent. We recall that two Cayley representations of the same multigraph are said to be equivalent if there exists a permutation on the vertex-set of the multigraph that induces an isomorphism from the group r to the group r' and sends S onto S'. Two Cayley representations (r, S), (r, S') are equivalent if and only if there exists an automorphism a of the group r that sends S onto S' (see [14]). The automorphism a is called a Cl-isomorphism (CI stands for Cayley Isomorphism). Adam [1] considered this problem for circulant graphs and formulated a well-known conjecture which was disproved in [9]. He conjectured that two circulant graphs Cir(n; S), Cir(n; S') are isomorphic if and only if there exists an integer m' G Zn, gcd(m', n) = 1, such that S' = {m'x : x G S}. Even though the conjecture was disproved, there are some circulant graphs for which it holds (see for instance [16]). In [7] the problem is studied for Cayley multigraphs of degree 4 which are associated to abelian groups. The results in [7] are described in terms of Adam isomorphisms. An Adam isomorphism from Cay(r, S) to Cay(r', S') is an isomorphism obtained from a permutation on the vertex-set of Cay(r, S), that makes (r, S), (r', S') equivalent, and an automorphism of the graph Cay(r', S'). By the definition of equivalent Cayley representations, the existence of an Adam isomorphism means that the groups r, r' are isomorphic and there exists an isomorphism between the groups that sends S onto S'. An Adam isomorphism between Cay(r, S) and Cay(r, S') is a Cl-isomorphism. Since the graphs X(s, t, r) can be represented as Cayley multigraphs, we can extend the notion of Adam isomorphism to the graphs X(s, t, r). We will say that the graphs X(s, t, r), X(s', t', r') are aAdam isomorphic if the corresponding Cayley multigraphs Cay(r, {±71, ±72}), Cay(r', {±7', ±y2}), respectively, are Adam isomorphic (Cay(r, {±y1, ±y2}), Cay(r', {±y', ±y2}) are described in Proposition 4.3 or 4.6). The following statements hold. Proposition 4.9. Every Adam isomorphism between the graphs X(s,t, r), X(s',t',r') sends the fundamental 2-factorization of X (s, t, r) onto the fundamental 2-factorization of X (s',t',r'). Proof. An Adam isomorphism between the graphs Cay(r, {±y1, ±y2}), Cay(r', {±y1 , ±y2}) sends a Yi-edge, i = 1, 2, of Cay(r, {±y1, ±Y2}) onto a t(Yi)-edge of Cay(r', {±Y1, ±y2}), where t(Yi) G {±y1 , ±y2}. Since Proposition 4.8 holds, every Adam isomorphism sends the red (respectively, the blue) 2-factor of X(s, t, r) onto the red (respectively, the blue) 2-factor of X(s', t', r') or vice versa. □ Proposition 4.10. Let s,t > 1, 0 < r < s - 1 and gcd(s, t, r) = d1. If r = 0, then there exists an integer k, 0 < k < st/d1, such that gcd(k, t) = 1 and k = r/d1 (mod s/d1). The graphs X (s, t, r), with r = 0, and X (s', t', r') are Adam isomorphic if and only if s' = s, t' = t, r' = s — r or s' = st/ gcd(s, r), t' = gcd(s, r) and r' = ±t(kd1/ gcd(s, r))-1 (mod st/ gcd(s, r)). The graphs X (s, t, 0), and X (s', t', r') are Adam isomorphic if and only if s' = t, t' = s and r' = 0 (mod t). Proof. We prove the assertion for s > 3, t > 2 and r = 0. The graph X(s,t,r) is the Cayley graph Cay(Zsi/dl x Zdl, {±(t/d1,1), ±(k, 0)}), since Proposition 4.6 holds. By Proposition 4.3 or 4.6, we can represent X(s',t',r') as the Cayley multigraph Cay(r', {±Y1, ±y2}). The graphs X(s, t, r), X(s', t', r') are Adam isomorphic if and only if there exists an isomorphism t between the groups Zst/dl x Zdl and r' that sends the set {±(t/d1,1), ±(k, 0)} onto the set {±y1 ,±y2}. Without loss of generality, we can set 12 Ars Math. Contemp. 12 (2017) 25-29 {±Yi} = {±t((t/d i, 1))} and {±y2}= {±t((k, 0))}. By the existence of t we can identify the group r' with the group Zst/dl x Zdl. Hence y' and y2 are elements of Zst/dl x Zdl of order s and st/ gcd(s,r), respectively, since (t/d i, 1) and (k, 0) have order s and st/ gcd(s, r), respectively (see the proof of Proposition 4.6 and 4.7). It is an easy matter to prove that an element (a, b) G Zst/dl x Zdl has order o(a) • o(b)/ gcd(o(a), o(b)) = st/(d i gcd(st/d i, a)), since di is a divisor of s and t. Hence (a, b) has order s if and only if gcd(st/di,a) = t/d i, that is, (a, b) = (m't/d i,b) where m' G Zst/dl, gcd(m', s) = 1, b is an arbitrary element of Zdl. The element (a, b) has order st/gcd(s, r) if and only if gcd(st/d i,a) = gcd(s,r)/di = gcd(s/di,k), since k is coprime with t and k = r/d i (mod s/di). Hence Cay(r', {±y', ±y2}) is a graph of type Cay(Zst/dl x Zdl, {±(m't/di, b), ±(a, b')}), where gcd(m', s) = 1, gcd(st/di,a) = gcd(s/d i,k), b and b' are suitable elements of Zdl. Note that a is coprime with t and the relation ta = rm't/d i (mod st/d i) holds, since t is an isomorphism and tk = rt/d i (mod st/d i). If we apply Proposition 4.2 to the graph Gi =Cay(Zst/dl x Zdl, {±(m't/di, b), ±(a, b')}) by setting y i = (m't/d i,b) (or 7 i = —(m't/di, b)), then G i can be represented as the graph X(s,t,r) or X(s,t, s — r). The graph X(s,t, r) is isomorphic to the graph G2 = X(s', t', r'), where s' = st/ gcd(s, r), t' = gcd(s, r), r' = ±t(kd i/ gcd(s, r))- i (mod st/ gcd(s,r)), since Proposition 4.7 holds. Hence G i is isomorphic to G2. The isomorphism between G i and G2 can be obtained also by applying Proposition 4.7 to the graph G i. For the remaining values of s, t, r we represent the graph X(s,t, r) as the Cayley multigraph in Proposition 4.3 and apply the previous method. □ The results that follow are based on the following theorem of [7]. Theorem 4.11 ([7]). Any two finite isomorphic connected undirected Cayley multigraphs of degree 4 coming from abelian groups are Adam isomorphic, unless they are obtained with the groups and sets Z4n, {±1, ±(2n — 1)} and Z2n x Z2, {±(1,0), ±(1,1)}. By Theorem 4.11 the existence of an isomorphism between two Cayley multigraphs of degree 4, that are associated to abelian groups, implies the existence of an Adam isomorphism, unless they are the graphs Cir(4n; ±1, ±(2n — 1)) and Cay(Z2n x Z2, {±(1,0), ±(1,1)}). The following statements are consequences of Theorem 4.11. Corollary 4.12. The graphs X (4n, 1,2n — 1) and X (s', t', r') are isomorphic if and only if s' = 4n, t' = 1, r' = 2n +1 or s' = 2n, t' = 2, r' G {2, 2n — 2}. Moreover, there is no isomorphism between X (4n, 1,2n — 1) and X (2n, 2,2) that sends the fundamental 2-factorization of X (4n, 1, 2n — 1) onto the fundamental 2-factorization of X (2n, 2, 2). Proof. The graph X(4n, 1,2n — 1) is the graph Cir(4n; ±1, ±(2n — 1)) (see Proposition 4.3). By Theorem 4.11, the graphs X(4n, 1, 2n — 1) and X(s', t', r') could be Adam isomorphic or not. If they are Adam isomorphic, then s' = 4n, t' = 1, r' = 2n +1, since Proposition 4.10 holds. If they are not Adam isomorphic, then X(s',t',r') is the graph Cay(Z2n x Z2, {±(1,0), ±(1,1)}) (see Theorem 4.11). Hence s' = 2n, t' = 2, r' G {2, 2n — 2} (see Proposition 4.6 and 4.10). The fundamental 2-factorization of X (4n, 1,2n—1) consists of two Hamiltonian cycles, whereas the fundamental 2-factorization of X(2n, 2, 2) consists of two 2-factors whose connected components are two 2n-cycles (see Proposition 4.8). Therefore no isomorphism between X(4n, 1, 2n—1) and X (2n, 2,2) can send the fundamental 2-factorization of X (4n, 1, 2n — 1) onto the fundamental 2-factorization of X(2n, 2, 2). □ S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 13 Proposition 4.13. Let X(s,t,r), X(s',t',r') be non-isomorphic to X(4n, 1, 2n - 1), X (2n, 2,2). Then X (s, t, r) and X (s', t', r') are isomorphic if and only if they are Adam isomorphic, that is, if and only if the parameters s', t', r' satisfy Proposition 4.10. Proof. The assertion follows from Theorem 4.11, and Proposition 4.10. □ 5 Special cubic graphs arising from X (s, t, r) graphs When we consider graphs X(s,t,r) we assume we are given a fundamental 2-factorization. This, in turn, implies we may turn the graph X(s, t, r) into a cubic one by appropriately splitting each vertex. We note in passing that the operation of vertex-splitting and its converse were successfully used in a different context in [20]. There are two complementary possibilities. Either X(s, t, r) arises from an I-graph or not. We consider each case separately. 5.1 I-graphs arising from X(s,t,r) In Theorem 3.3 we remarked that any special cubic graph with a blue and red 2-factoriza-tion gives rise to the associated quartic graph with a blue and red 2-factorization. In Lemma 3.5, we showed that a proper I-graph I(n,p, q) is special and gives rise to the associated circulant graph Q(n, p, q). The following holds. Lemma 5.1. The circulant graph Cir(n; p, q) = Q(n,p, q) arising from a connected I-graph I(n,p,q) by contracting the spokes is the graph X(s,t,r) with t = gcd(n, q), s = n/t > 3 and r = ±p(q/t)-1 (mod s). Proof. The result follows from Proposition 4.2 by setting r = Zn, y1 = q, y2 = p. Whence tp = rq for some integer r, 0 < r < s — 1, that is, r is a solution to the equation r(q/t) = p (mod s). By the Chinese Remainder Theorem, r = p(q/t)-1 (mod s). □ Theorem 5.2. The graph X (s,t,r) arises from a connected I-graph by contracting the spokes if and only if gcd(s, t, r) = 1 and (t, r)= (2,0) for odd values of s. In this case, the graph X (s, t, r) together with its fundamental 2-factorization, is in one-to-one correspondence with the I-graph I(st, k,t), where 0 < k < st, gcd(k,t) = 1 and k = r (mod s) (in particular, k = s if r = 0). If at least one of the integers k, t, gcd(s, r) is 1, then X(s, t, r) corresponds to a generalized Petersen graph. Proof. Assume that X(s, t, r) arises from the connected I-graph I(n,p, q) by contracting the spokes. By Lemma 5.1, t = gcd(n, q), s = n/t > 3 and r(q/t) = p (mod s). Whence (t, r) = (2,0) if s is odd, otherwise p = 0 (which is not possible). We show that gcd(s, t, r) = 1. Suppose, on the contrary, that gcd(s, t, r) = d1 = 1, then d1 is a divisor of gcd(t,p) since r(q/t) = p (mod s). That yields a contradiction, since gcd(t,p) = 1 (see Proposition 3.4). Hence gcd(s, t,r) = 1. Assume that gcd(s, t, r) = 1. We show that X(s, t, r) arises from a connected I-graph by contracting the spokes. Since gcd(s, t, r) = 1, the graph X(s,t,r) can be represented as the circulant graph Cir(st; ±t, ±k), where 0 < k < st, gcd(t, k) = 1 and k = r (mod s) (see Proposition 4.6). If r = 0, then we can set k = s, since Proposition 4.3 holds. The graph I (st, k, t) is connected and it gives rise to the graph X (s, t, r), since Lemma 5.1 holds. By Theorem 3.3, the graph X(s, t, r), together with its fundamental 2-factorization, is in one-to-one correspondence with the I-graph I(st,k,t). If k = 1 or t = 1, then 14 Ars Math. Contemp. 12 (2017) 25-29 X(s, t, r) corresponds to a generalized Petersen graph (see [3]). If gcd(s,r) = 1 then X(s, t, r) is isomorphic to X(st, 1, r') (see Proposition 4.10. By the previous remarks, the graph X(st, 1, r') corresponds to a generalized Petersen graph. The assertion follows. □ It is straightforward to see that isomorphic X(s, t, r) graphs give rise to isomorphic I-graphs and also the converse is true. By Corollary 4.12 and Proposition 4.13, the circulant graphs X(s,t, r), X(s',t', r') are isomorphic if and only if they are Adam-isomorphic, that is, there exists an automorphism of the cyclic group of order st = s't' that sends the defining set of the circulant graph X(s, t, r) onto the defining set of the circulant graph X(s',t',r'). This fact is equivalent to the results proved in [13] about the isomorphism between I-graphs. 5.2 Special Generalized I-graphs In this section we consider the special cubic graphs that correspond to the graphs X(s, t, r) with gcd(s, t, r) = 1, according to the correspondence described in Theorem 3.3. By Proposition 5.2, these special cubic graphs do not belong to the family of connected I-graphs. By Theorem 3.3 and Definition 4.1, we can define a family of special cubic graphs containing the family of connected I-graphs as a subfamily. We call this family Special Generalized I-graphs. This family is not contained in the family of GI-graphs [6]. Let s > 1, t > 1 and 0 < r < s — 1. We define a Special Generalized I-graph SGI (st, s, t, r) as a cubic graph of order st with vertex-set V = {ui,j , «i,j : 0 < i < t — 1,0 < j < s — 1} and edge-set E = {[«¿j ,ui,j+1], [«i,j, «i j ] : 0 < i < t — 1,0 < j < s — 1}U{[ui,j, «i+i,j ] : 0 < i < t — 2, 0 < j < s - 1}U{K-i,j, uj ] : 0 < j < s — 1} (the addition j + 1 and j+r are considered modulo s). For s =1 or (t, r) = (1,0), a special generalized I-graph has loops. For s = 2 or (t, r) = (2,0), it has multiple edges. For the other values of s, t, r, it is a simple cubic graph. We say that a vertex uijj (respectively, «j) is an outer vertex (respectively, an inner vertex). We say that an edge [ui,j, uiij+1] (respectively, [«i,j, «i+1,j]) is an outer edge (respectively, an inner edge). We say that an edge [ui,j ,«i j] is a spoke. The spokes constitute the special 1-factor. The graph arising from SGI (st, s,t,r) by contracting the spokes is the graph X (s,t,r). The horizontal edges of X (s, t, r) correspond to the outer edges of SGI (st, s, t, r), vertical and diagonal edges of X (s, t, r) correspond to the inner edges of SGI (st, s, t, r). A generalization of the proof of Proposition 5.2 gives the following statement. Proposition 5.3. Let s > 1, t > 1, 0 < r < s — 1 and d1 = gcd(s,t,r). The graph X (s,t,r), together with its fundamental 2-factorization, is in one-to-one correspondence with the graph SGI (st, s,t, k) where k = s if r = 0, otherwise 0 < k < st/d1, gcd(k,t) = 1 and k = r/d1 (mod s/d1). By Corollary 4.12, the graphs X(4n, 1, 2n — 1) and X(2n, 2, 2) are isomorphic, but no isomorphism between them sends the fundamental 2-factorization of X (4n, 1, 2n — 1) onto the fundamental 2-factorization of X(2n, 2, 2). This fact means that the application of Theorem 3.3 to the graphs X(4n, 1,2n — 1) and X(2n, 2,2) yields non-isomorphic special cubic graphs. As a matter of fact, Proposition 5.2 says that the graph X(4n, 1,2n — 1) is in one-to-one correspondence with a connected I-graph, whereas X(2n, 2, 2) does not correspond to any I-graph. For instance, for n = 2 the graph X(8,1, 3) is associated with the Mobius-Kantor graph of girth 6, [15, 17], while X(4,2, 2) arises from a graph of girth 4 (see Figure 2). S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 15 Figure 2: The cubic split X(4, 2, 2) graph is SGI(8,4,2,2). The thick edges represent the special 1-factor. 6 Good Eulerian tours in X (s, t, r) graphs In this section we construct good Eulerian subgraphs of X(s, t, r). For each X(s, t, r) we denote by W(s, t, r) the constructed good Eulerian subgraph. By Proposition 3.2, a spanning Eulerian subgraph W of X(s, t, r) is admissible if and only if at each 2-valent vertex exactly one edge is horizontal. We consider X(s, t, r) being embedded into the torus with quadrilateral faces. Hence any of its subgraphs may be viewed embedded in the same surface. A tour in W may be regarded as as a straight-ahead walk (or SAW) on the surface [18]. A good Eulerian tour of W is an Eulerian SAW that uses only allowed transitions, that is, the tour cannot switch from a horizontal to a vertical (or diagonal) edge when it visits a 4-valent vertex of W. For instance, the graph W in Figure 4(a) is an admissible subgraph of XX(5, 4, 3); the tour E == (xq,xq,xi, X2, xg, x^, xq , x 3, x 1, x 1, x 1, X2,x2, X2, X2, x2, xQ, xQ,xf, x2, x§, x2, x2, x2, x§, xg, xq, x4, x°) is a good Eulerian SAW of W; hence W is a good Eulerian subgraph of X(5,4,3). If we delete the diagonal edges in X(s, t, r), we obtain a spanning subgraph that we denote by X'(s, t, r). Clearly X'(s, t, r) is the cartesian product of a cycle Cs with a path Pt embedded in the torus or cylinder. If we further delete an edge in Cs we obtain a path Ps. We denote the cartesian product of Ps and Pt by X''(s,t,r) and obtain a spanning subgraph of X'(s, t, r) and X(s, t, r). In order to simplify the constructions, we will seek to find good Eulerian subgraphs in X'(s, t, r) or in X''(s, t, r). In this case the resulting good Eulerian subgraph will be denoted by W'(s, t, r) and W''(s, t, r), respectively. This simplification makes sense, since neither X'(s, t, r) nor X''(s, t, r) depend on the parameter r. Hence any Eulerian subgraph W'(s, t, r) or W''(s, t, r) is good for any r. 6.1 Method of construction We give some lemmas that will be used in the construction of a good Eulerian subgraph W(s, t, r). Given a graph X(s, t, r), for every row index i, 0 < i < t - 1, we denote by V the set of vertical edges V = {[xj, x j+1 ] : 0 < j < s - 1}. For every column index j, 0 < j < s - 1, we denote by Hj the set of horizontal edges Hj = {[xj, xj+1] : 0 < i < t - 1}. Let H be a subgraph of X(s, t, r). We say that H can be expanded vertically (from row 16 Ars Math. Contemp. 12 (2017) 25-29 i) if |E(H) n Vi | = s - 1 or s - 2 > 0 (for s = 3 we require |E(H) n Vi| = 2). We say that H can be expanded horizontally (from column j) if |E(H) n Hj t - 1 or t - 2 > 0 (for t — 3 we require |E(H) n Hj | = 2). The following statements hold. Lemma 6.1. Let W(s, ti, r) be a good Eulerian subgraph that can be expanded vertically. Then there exists a good Eulerian subgraph W(s,t, r) for every t > t1, t = t1 (mod 2). Proof. We use the graph Wi — W (s,ti,r) to construct a good Eulerian subgraph W(s, t, r). By the assumptions, |E(W1) n Vi| — s - 1 or s - 2 for some row index i, 0 < i < t - 1. By the symmetry properties of the graph X(s,t1 ,r), we can cyclically permute its rows so that we can assume 0 < i < t - 1. We treat separately the cases |E(Wi) n Vj| — s - 1 and |E(Wi) n V| — s - 2. Consider |E(Wi) n Vi| — s - 1 and denote by [x^, xa+1] the vertical edge of Vj which is missing in W1. We can cyclically permute the columns of X(s, t1, r) and assume a — 0. We subdivide every vertical edge [xj, xj+1], with 0 < j < s -1, by inserting two new vertices, namely, yj and yj+1 such that yj is adjacent to xj and yj+1 is adjacent to xj+1, and we add two new vertices yj,1, y0 between x0+1 and x0 in column 0. We now delete the edge [yj _ 1, yj-11 ] and replace it with the path from yj_1 to yj_i composed of the edges [yj+1, yj+i], [yj, yj+1], 0 < j < s - 2, and [y0, y0+1]. The resulting graph is a good Eulerian subgraph W(s,t1 + 2, r). We can iterate the process and find a good Eulerian subgraph W(s, t, r) for every t > t1, t = t1 (mod 2). The case | E(W1) n Vj | — s - 2 can be treated analogously to the case | E( W1) n Vj | — s -1. As an example, consider the graph W"(6,5, r) in Figure 3. It can be expanded vertically from row 1 and it yields a good Eulerian subgraph W"(6, 7, r). □ vertical expansion Figure 3: A vertical expansion of the good Eulerian subgraph W'(6,5, r) yields a good Eulerian subgraph W(6, 7, r). In the following lemma we consider horizontal expansions. In this case we have to pay attention to the diagonal edges of W(s, t, r), if any exist. If [xj-1, x0+r], where j + r is considered modulo s, is a diagonal edge of W(s, t, r), then we can assume j < j + r, since we can cyclically permute the columns of W(s, t, r). Therefore we can say that a diagonal edge [xj , x0+r ] crosses column I if j < I < j + r. Lemma 6.2. Let W(s1, t, r1) be a good Eulerian subgraph that can be expanded horizontally from column L If no diagonal edge of W (s1, t, r1) crosses column £, then there exists a good Eulerian subgraph W (s,t, r1) for every s > s1, s = s1 (mod 2). If every diagonal S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 17 edge crosses column i, then there exists a good Eulerian subgraph W(si + r — ri, t, r) for every r > r1, r = r1 (mod 2). Proof. We apply the method described in Lemma 6.1 to the edges in H^. If every diagonal edge of W(s1, t, r1) crosses column i, then by subdividing the edges of H we can shift of r — r1 steps the diagonal edges of W (s1, t, r1). If no diagonal edge of W (s1, t, r1) crosses column i, then no diagonal edge is shifted. As an example, consider the graph W(5,4, 3) in Figure 4. If we expand horizontally the graph from column i = 0, then no diagonal edge crosses column i and we obtain a good Eulerian subgraph W(7,4,3). If we expand horizontally the graph from column i = 2, then every diagonal edge crosses column i and we obtain a good Eulerian subgraph W(7,4,5). □ Figure 4: A good Eulerian subgraph: (a) W(5,4, 3); (b) W(7,4,3); (c) W(7,4,5). The graphs W(7,4,3) and W(7,4,5) are obtained from W(5,4, 3) by an horizontal expansion from column 0 and column 2, respectively. 6.2 Constructions of good Eulerian subgraphs. We apply the lemmas described in Section 6.1 to construct a good Eulerian subgraph W(s, t, r). It is straightforward to see that the existence of loops in X(s, t, r) excludes the existence of a good Eulerian subgraph W(1, t, r) and W(s, 1,0). Analogously, the existence of horizontal parallel edges in X(2, t, r) excludes the existence of a good Eulerian subgraph W(2, t, r) with t odd and W(2, t, 1) with t even, t > 2, (see Case 2 in the proof of Lemma 6.5 for a good Eulerian subgraph W (2,2,1) and W (2, t, 0) with t even). Hence we can consider s > 3 and (t, r)= (1,0). The following hold. Proposition 6.3. The graph X (s, 1, r), r = 0, possesses a good Eulerian subgraph, unless s = 6m + 5, with m > 0, and r G {2, s - 2, (s + 1)/2, (s - 1)/2}. Proof. By Proposition 4.3, the graph X(s, 1, r) can be represented as the circulant multigraph Cir(st; ±1, ±r). By Proposition 5.2, the graph X(s, 1, r) corresponds to the generalized Petersen graph I(s, r, 1) or G(s, r). In particular, the graph X(6m + 5,1, 2) corresponds to the generalized Petersen graph G(6m + 5,2). Hence X(s, 1, r) has a good Eulerian subgraph, unless it is isomorphic to X(6m + 5,1, 2), since Theorems 1.1 and 2.2 hold. By Proposition 4.13, the graphs that are isomorphic to X(6m+5,1,2) are X(6m+5,1, r'), where r' G {2, 6m + 3} or r' = ±2-1 (mod 6m + 5), that is, r' G {3m + 3, 3m + 2}, since r' < 6m + 5. □ We can construct a good Eulerian subgraph W(s, 1, r), r = 0, without using Theorem 1.1. More specifically, by Proposition 4.10 the graph X(s, 1, r), with r = 0, is isomorphic 18 Ars Math. Contemp. 12 (2017) 25-29 to the graph X(s/ gcd(s, r), gcd(s, r), r'), where r' = ±r-1 (mod s). For r = 0 and gcd(s, r) > 1, a construction of a good Eulerian subgraph can be found in the proof of Lemma 6.5. We can also provide an ad hoc construction for the case gcd(s, r) = 1, but we prefer to omit this construction, since the existence of a good Eulerian subgraph W(s, 1, r), r = 0, is known (see Proposition 6.3) and the construction is based on the method of Lemma 6.5. We will show that the graph X(6m + 5,1,2), m > 0, has no good Eulerian subgraph, that is, the generalized Petersen graph is not Hamiltonian. The following statement is a consequence of Proposition 6.3 and it will be used in the proof of Lemma 6.5. Proposition 6.4. The graph X(s,t, r), with s > 3, t > 1 and gcd(s,r) = 1 has a good Eulerian subgraph. Proof. By Proposition 4.6, the graph X(s, t, r) can be represented as the circulant graph C«r(st; ±t, ±k), where gcd(k, t) = 1 and k = r (mod s). By Proposition 4.10, the graph X(s,t,r) is isomorphic to the graph X(st, 1,r'), with r' = 0, since gcd(s,r) = 1. If st ^ 5 (mod 6), then the assertion follows from Proposition 6.3 (see Proposition 4.10). Consider st = 5 (mod 6). We show that X(s, t, r) is not isomorphic to X(6m + 5,1, 2), m > 0. Suppose, on the contrary, that X(s, t, r) is isomorphic to X(6m + 5,1, 2). Then X(st, 1, r')= X(6m + 5,1,r'), where r' G {2, st - 2, (st + 1)/2, (st - 1)/2} (see Proposition 6.3). By Proposition 4.10, the integer r' satisfies the relation r' = ±tk-1 (mod st). Whence t is a divisor of r'. That yields a contradiction, since r' G {2, st — 2, (st + 1)/2, (st — 1)/2} and t is coprime with the integers in {2, st — 2, (st + 1)/2, (st — 1)/2}. □ Lemma 6.5. Let s > 3, t > 2 and 0 < r < s — 1. There exists a good Eulerian subgraph W(s, t, r), unless s is odd and (t, r) = (2, 0). Proof. We treat separately the cases: t = 3; s, t even; s even, t odd, t > 5; s odd, t even; s, t odd, t > 5. When we will speak of "vertical" and "horizontal" expansion we refer implicitly to Lemma 6.1 and 6.2, respectively. Case 1: t = 3. This case is treated in Section 8, since it requires a lengthy description. (a) (b) (c) (d) Figure 5: A good Eulerian subgraph: (a) W'(2,2, r); (b) W''(4,4, r); (c) W'(6, 6, r); (d) W''(6, 8, r). S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 19 Case 2: s even, t even. The graph W''(6,8, r) in Figure 5(d) can be expanded vertically from row 1 and horizontally from column 2. It yields a good Eulerian subgraph W''(s, t, r) for every s, t even s > 6, t > 8. It remains to construct a good Eulerian subgraph W"(s,t,r) for s > 6, t = 2, 4, 6 and W"(4,t,r) for t > 2, t even. The graph W'(2,2, r) in Figure 5(a) can be expanded horizontally from column 0 or 1. It yields a good Eulerian subgraph W'(s, 2, r) for every s even, s > 2. We expand horizontally the graph W''(4,4, r) in Figure 5(b) and obtain W''(s, 4, r) for every s even, s > 4. We rotate W''(s, 4, r) by 90 degrees clockwise (around a vertex) and obtain a good Eulerian subgraph W''(4, t, r) for every t even, t > 4. We expand horizontally the graph W''(6,6, r) in Figure 5(c) from column 3 and obtain W''(s, 6, r) for every s even, s > 6. Case 3: s even, t odd, t > 5. The graph W'(6, 5, r) in Figure 3 can be expanded vertically from row 2 and horizontally from column 3. It yields a good Eulerian subgraph W'(s, t, r) for every s even, s > 6, t odd, t > 5. It remains to construct W(4, t, r) with t odd, t > 5, 0 < r < 3. Since X(4, t, r) is isomorphic to X(4, t, 4 — r), we can consider 0 < r < 2. A good Eulerian subgraph for W(4, t, 0), t odd, t > 5, can be obtained from W(4, 3,0) in Figure 6(a) by a vertical expansion from row 1. The existence of a good Eulerian subgraph W(4,t, 1) follows from Proposition 6.4. By Proposition 4.10, the graph X(4, t, 2) is isomorphic to the graph X(2t, 2, r'). By the results in Case 2, there exists a good Eulerian subgraph W(2t, 2, r'). (a) (b) (c) Figure 6: A good Eulerian subgraph: (a) W(3, 3,0); (b) W(4, 3,0); (c) W(6, 3,0). Case 4: s odd, t even. By Proposition 4.10, the graph X(s, t, r), with r = 0, is isomorphic to the graph X(st/ gcd(s,r), gcd(s, r), r'), with r' = 0, or to X(t, s, 0) if r = 0. If r = 0 and gcd(s, r) = 1 or 3, then the existence of a good Eulerian subgraph follows from Proposition 6.4 or from the results in Case 1, respectively. Note that st/ gcd(s, r) > 4, since t is even and 0 < r = s — 1. Hence, for gcd(s, r) > 5, the existence of a good Eulerian subgraph follows from Case 3. Consider r = 0. There is no good Eulerian subgraph W(s, 2,0), because of the existence of parallel vertical edges. Consider t > 4. As remarked, the graph X(s,t, 0) is isomorphic to the graph X(t, s, 0). For s > 5 the existence of a good Eulerian subgraph W (t, s, 0) follows from the results in Case 3. The existence of a good Eulerian subgraph W(t, 3,0) follows from Case 1. Case 5: s odd, t odd, t > 5. A good Eulerian subgraph W(s, t, 0) can be obtained from the graph W(3,3,0) in Figure 6(a). If r G {1, 2}, then the existence of a good Eulerian subgraph follows from Proposition 6.4. Consider 3 < r < s — 3 and s > 7. Since X(s, t, r) is isomorphic to X(s, t, s — r) and s is odd, we can construct 20 Ars Math. Contemp. 12 (2017) 25-29 a good Eulerian subgraph W(s, t, r) for every s, r odd, s > 7, 3 < r < s - 4. The graph W(7,5,3) in Figure 10(c) can be expanded horizontally from column 4 and vertically from row 1 (or 2). It yields a good Eulerian subgraph W(s, t, 3) for every s, t odd, s > 7, t > 5. Since s - r + 3 > 7, we can consider the graph W(s - r + 3, t, 3) arising from W(7, 5,3) in Figure 10(c). We expand horizontally the graph W(s - r + 3,t, 3) from column 2 and obtain a good Eulerian subgraph W(s, t, r) for every s, t, r odd, s > 7, t > 5 and 3 < r < s - 4. □ Proposition 6.6. The graph X(6m + 5,1, 2), m > 0, has no good Eulerian subgraph. Consequently, the generalized Petersen graph G(6m + 5,2) has no Hamiltonian cycle. Proof. We give a sketch of the proof by showing that X (5,1, 2) has no good Eulerian subgraph. Suppose, on the contrary, that W is a good Eulerian subgraph of X(6m + 5,1, 2). Since the unique horizontal layer of W has an odd number of vertices, the graph W contains at least one path P2j+1 consisting of 2j horizontal edges. It is possible to prove that 2j = 2 (if 2j > 2, then W is not good). Without loss of generality we can set P2j+1 = (x0, xi, x0). Whence [x0, x°] G E( W) and no other horizontal edge of X(5,1, 2) belongs to E(W). Moreover, [xi, x0],[xi, x0] are edges of W, since W is admissible and xi is 4-valent in W. Whence [x0,x0] G E(W) and each admissible tour of W contains the component A = (x0, x0, x0, x0). That yields a contradiction, since A is not a spanning subgraph of X(6m + 5,1, 2). Hence X(5,1, 2) has no good Eulerian subgraph. By Theorem 2.2, the graph G(5, 2) has no Hamiltonian cycle. The proof can be generalized to the case G(6m + 5, 2) with m > 0. □ 7 Characterization of Hamiltonian I-graphs Now we are ready to prove the main theorem. Proof of Theorem 1.2. By Theorem 1.1, a generalized Petersen graph is Hamiltonian if and only if it is not isomorphic to G(6m + 5,2), m > 0. We prove that a proper I-graph is Hamiltonian. By Lemma 3.5, a proper I-graph I(n,p, q) is special and its associated quartic graph X is the circulant graph Cir(n; p, q). By Lemma 5.1, the graph Cir(n;p, q) can be represented as the graph X(s,t,r), where t = gcd(n, q), s = n/t > 3, r = ±p(q/t)-1 (mod s) and (t,r)= (2,0) for odd values of s. By Lemma 6.5, the graph X(s, t, r) has a good Eulerian subgraph. The assertion follows from Theorem 2.2. □ By Theorem 2.2 and Lemma 6.5, we can extend the result of Theorem 1.2, about the existence of a Hamiltonian cycle, to the special generalized I-graphs. As a consequence of Theorem 1.2, a proper I-graph is 3-edge-colorable or, equiva-lently, 1-factorizable (because it is cubic and Hamiltonian). A widely studied property for 1-factorizable graphs is the property of admitting a perfect 1-factorization. We recall that a 1-factorization is perfect if the union of any pair of distinct 1-factors is a Hamiltonian cycle. Partial results are known for generalized Petersen graphs: G(n, k) admits a perfect 1-factorization when (n, k) = (3,1); (n, k) = (n, 2) with n = 3,4 (mod 6); (n, k) = (9, 3); (n, k) = (3d, d) with d odd; (n, k) = (3d, k) with k > 1, d odd, 3d and k coprime (see [4]). So, it is quite natural to extend the same problem to proper I-graphs. S. Bonvicini and T. Pisanski: A novel characterization of cubic Hamiltonian graphs 21 Some further problems can be considered: the generalization of the existence of good Eulerian tour to other graph bundles of a cycle over a cycle, the characterization of Hamiltonian GI-graphs or of Hamilton-laceable I-graphs. In [8], the authors proved by a computer search that all bipartite connected I-graphs on 2n < 200 vertices are Hamilton-laceable. 8 Appendix. Proof of Lemma 6.5 Case 1, t = 3. We expand horizontally the graph W(3,3,0) in Figure 6(a) from column 0 and obtain a good Eulerian subgraph W(s, 3,0) for every s odd, s > 3. A good Eulerian subgraph W(s, 3,0) with s even can be obtained from the graphs W(4, 3,0) and W(6, 3,0) in Figure 6(b)-(c). As an example, the graph W(8,3,0) in Figure 7(a) has been obtained by connecting two copies of the graph W(4,3,0). The graph W(10,3,0) in Figure 7(b) has been obtained by connecting the graphs W(4, 3,0) and W(6, 3,0). For r = 1 the existence of a good Eulerian subgraph W(s, 3,1) follows from Proposition 6.4. Hence we can consider 2 < r < s/2, since X(s, 3,r) is isomorphic to X(s, 3, s — r). The graph W(4, 3, 2) in Figure 7(c) can be expanded horizontally from column 3. It yields a good Eulerian subgraph W(s, 3, 2) for every s even, s > 4. Since s — r + 2 > 4, we can consider the graph W(s — r + 2,3, 2) obtained from W(4, 3, 2) in Figure 7(c). We expand horizontally W(s — r + 2,3, r) from column 1 and obtain a good Eulerian subgraph W(s, 3, r) for every s, r even, s > 4, 2 < r < s/2. Analogously, the graphs W(6,3, 3), W(8,3,3) and W(10, 3,5) in Figure 8 yield a good Eulerian subgraph W(s, 3, r) for every s even, r odd, 3 < r < s/2. More specifically, we expand horizontally the graph W(8, 3, 3) from column 7 and obtain a good Eulerian subgraph W(s, 3, 3) for every even integer s > 8. The graph W(10,3, 5) can be expanded horizontally from column 9 (or 0). It yields a good Eulerian subgraph W(s, 3,5) for every even integer s, s > 10. Since s — r + 5 > 10, we can consider the graph W(s — r + 5, 3, 5) obtained from W(10,3,5) in Figure 8(c). We expand W(s — r + 5,3,5) from column 4 and obtain a good Eulerian subgraph W(s, 3, r) for every s even, s > 10, r odd, 5 < r < s/2. Figure 8: A good Eulerian subgraph: (a) W(6,3,3); (b) W(8,3, 3); (c) W(10,3,5). Consider s odd, s > 5. The graph W(5, 3, 2) in Figure 9(a) can be expanded horizontally from column 4. It yields a good Eulerian subgraph W(s, 3,2) for every s odd, 22 Ars Math. Contemp. 12 (2017) 25-29 s > 5. Analogously, the graph W(9,3,4) in Figure 9(b) yields a good Eulerian subgraph W(s, 3,4) for every s odd, s > 9. The graph W(13,3, 6) in Figure 9(c) can be expanded horizontally from column 2 and column 10. It yields a good Eulerian subgraph W(2r + 1, 3, r) with r even, 6 < r < s/2. Since s — 2r + 1 > 0, we can expand W(2r + 1, 3, r) from column 2r and find a good Eulerian subgraph W(s, 3, r) for every s odd, s > 13, r even, r > 6. It remains to construct a good Eulerian subgraph W(s, 3, r) with s, r odd, s > 7, 3 < r < s/2. We use the graph W(7, 3,3) in Figure 10(a) to construct a good Eulerian subgraph W(2r + 1, 3, r) with r odd, r > 3. As an example, the graph W(11,3, 5) in Figure 10(b) has been obtained by expanding horizontally the graph W(7,3, 3) from column r = 3 and s —1=6 and by adding new diagonal edges. If we iterate the process, then we obtain a good Eulerian subgraph W(2r + 1,3, r) with r odd, r > 3. 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ARS MATHEMATICA CONTEMPORANEA 12 (2017) 25-29 The parameters of Fibonacci and Lucas cubes* Marko Milosevic Faculty of Sciences and Mathematics, University ofNis, Serbia Received 10 August 2015, accepted 27 December 2015, published online 21 March 2016 Motivated by the conjectures from Castro, et al. in 2011, in this paper we use integer programming formulations for computing the domination number, the 2-packing number and the independent domination number of Fibonacci cubes and Lucas cubes for n < 13. Keywords: Fibonacci cubes, Lucas cubes, domination number, 2-packing number. Math. Subj. Class.: 05C69, 05C25 1 Introduction Hypercubes form one of the most applicable classes of graphs with many appealing properties. The n-cube Qn is the graph whose vertices are all binary strings of length n, and two vertices are adjacent if they differ in exactly one position. The Fibonacci cubes were introduced as a model for interconnection networks [4, 2]. They offer challenging mathematical and computational problems, and admit a recursive decomposition into smaller Fibonacci cubes (see [5], [6], [8] for their structural properties). The Fibonacci cubes can be recognized in O(m log n) time (where n is the order and m the size of a given graph) [10]. The Lucas cubes [7] form a class of graphs closely related to the Fibonacci cubes, obtained by removing some vertices from the Fibonacci cubes. Let Qn be the n-dimensional hypercube. A Fibonacci string of length n is a binary string bib2.. .bn with bi • bi+1 =0 for 1 < i < n. In other words, Fibonacci strings are binary strings that contain no consecutive ones. The Fibonacci cube rn, for n > 1 is the subgraph of Qn induced by the Fibonacci strings of length n. A Fibonacci string bib2... bn is a Lucas string if bi • bn = 0. In other words, Lucas strings are binary strings *This work was supported by Research Grants 174010 and 174033 of Serbian Ministry of Education and Science. E-mail addresses: aleksandari@gmail.com (Aleksandar Ilic), marko643@gmail.com (Marko Milosevic) Aleksandar Ilic Facebook Inc, Menlo Park, CA, USA Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 26 Ars Math. Contemp. 12 (2017) 25-29 that contain no consecutive ones circularly. The Lucas cube An, for n > 1 is the subgraph of Qn induced by the Lucas strings of length n. It is well-known that |V(T„)| = Fn+2, where Fn are the Fibonacci numbers: F0 =0, F1 = 1, Fn+1 = Fn + Fn-1 for n > 1. Similarly, |V(An)| = Ln for n > 1, where Ln are the Lucas numbers: L0 = 2, L1 = 1, Ln+1 = Ln + Ln-i for n > 1. Let G be a graph. Set D C V(G) is a dominating set if every vertex from V(G) either belongs to D or is adjacent to some vertex from D. The domination number y(G) is the minimum cardinality of a dominating set of G. A set X C V (G) is called a 2-packing if d(u, v) > 2 for any two different vertices u and v of X. The 2-packing number p(G) is the maximum cardinality of a 2-packing of G. It is well-known that for any graph G holds Y(G) > p(G). An independent set or stable set is a set of vertices in a graph, no two of which are adjacent. The independent domination number ¿(G) of a graph G is the size of the smallest independent dominating set (or, equivalently, the size of the smallest maximal independent set). The minimum dominating set in a graph will not necessarily be independent, but the size of a minimum dominating set is always less than or equal to the size of a minimum maximal independent set, y(G) < ¿(G). Pike and Zou in [9] obtained a lower bound for the domination number of Fibonacci cube of order n and determined the exact value of the domination number of Fibonacci cubes of order at most 8. Castro et al. in [1] obtained upper and lower bounds for the domination and 2-packing number of Fibonacci and Lucas cubes. Furthermore, the authors obtained the exact values for Y(rn) and Y(An) for n < 9 and for p(rn) and p(An) for n < 10. In this paper we use integer programming method to compute the exact values of the domination, 2-packing and independent domination number of Fibonacci and Lucas cubes for n < 13, which resolves the conjecture from [1]. 2 Main results For each subset of the vertex set S ç V(G) define 1 if i e S 0 if i e V \ S. The neighborhood N(v) of a vertex v in a graph G is the induced subgraph of G consisting of all vertices adjacent to v and all edges connecting two such vertices. Let N [v] = N (v) U {v} denote the closed neighborhood of the vertex v. The domination number of G can be formulated as the following 0 -1 integer programming problem: n y(G) = min ^ Xi (2.1) i=i subject to ^ Xj > 1, (2.2) jeN [i] Xi e {0,1}, for all 1 < i < n. (2.3) It is easy to see that the conditions (2.2) and (2.3) define dominating set S and vice versa [3]. For Fibonacci cube rn this formulation has Fn+2 variables and 2Fn+2 constraints, A. Ilic & M. Milosevic: The parameters of Fibonacci and Lucas Cubes 27 while each condition from (2.2) contains at most n variables. For Lucas cube An this formulation has Ln variables and 2Ln constrains, while each condition from (2.2) contains at most n variables. The 2-packing number of G can be formulated as the following 0 - 1 integer programming problem: n p(G) = max X>i (2.4) i=i subject to ^ xj < 1, (2.5) jew [i] xi G {0,1}, for all 1 < i < n. (2.6) We will prove that the conditions (2.5) and (2.6) define 2-packing set S and vice versa. Let S be a 2-packing set. Since S does not contain two vertices on distance 1 or 2, for each v G V(G) there is at most one vertex from the closed neighborhood N[v] which belongs to S. Assume now that the set S satisfies the condition (2.5) and let u and v be two vertices from S on distance 2. In that case for the shortest path vwu, we have J2jeN[w] xj > 2, which is impossible. Therefore, S is a 2-packing set. The independent domination number G can be formulated as the following 0 -1 integer programming problem: n i(G) = min ^ xi (2.7) i=i subject to ^ xj > 1, (2.8) jew [i] (n - 1)xi + xj < n - 1, (2.9) jew (i) xi G {0,1}, for all 1 < i < n. (2.10) The conditions (2.8) and (2.10) define domination set S, while the condition (2.9) ensures the independence. For xi =0 we have always true (i) xj < n - 1, while for xi = 1 we have J2jeN(i) xj < 0 which is equivalent to 2jeN[i] xj = 1. This proves that the formulation is correct. For Fibonacci cube rn this formulation has Fn+2 variables and 3Fn+2 constraints, while each conditions from (2.8) and (2.9) contain at most n variables. For Lucas cube An this formulation has Ln variables and 3Ln constrains, while each condition from (2.8) and (2.9) contain at most n variables. The tests were performed on the Intel Core 2 Duo T5800 2.0 GHz with 2 GB RAM running the Linux operating system and using CPLEX 8.1. The results are summarized in Tables 1 and 2. In Tables 3 and 4 we give some examples of dominating sets and 2-packing sets that were obtained during the computation of these values. These results resolve the conjecture from [1] and support Problem 5.1 for n < 12. 28 Ars Math. Contemp. 12 (2017) 25-29 n 1 2 3 4 5 6 7 8 9 10 11 |V (r„)| 2 3 5 8 13 21 34 55 89 144 233 I e (r„)| 1 2 5 10 20 38 71 130 235 420 744 7(r„) 1 1 2 3 4 5 8 12 17 25 p(r„) 1 1 2 2 3 5 6 9 14 20 29 i(r„) 1 1 2 3 4 5 8 12 19 26 Table 1: Parameters of small Fibonacci cubes. n 1 2 3 4 5 6 7 8 9 10 11 12 |V(An) I 1 3 4 7 11 18 29 47 76 123 199 322 IE (An) | 0 2 3 8 15 30 56 104 189 340 605 1068 Y(An) 1 1 1 3 4 5 7 11 16 23 35 P(An) 1 1 1 2 3 5 6 8 13 18 26 38 i(An) 1 1 1 3 4 5 8 11 17 24 35 Table 2: Parameters of small Lucas cubes. Dominating set r(io) I A(ii) (0, 1, 0, 1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 1, 0, 0, 0, 0) (1 0, 1, 0, 1, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0) (1, 0, 1, 0, 0, 1, 0, 0, 0, 0), (1, 0, 0, 1, 0, 0, 1, 0, 0, 0) (0 1, 0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0) (0, 0, 0, 0, 1, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0, 0) (0 0, 1, 0, 1, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0) (1, 0, 1, 0, 1, 0, 0, 1, 0, 0), (1, 0, 0, 1, 0, 1, 0, 1, 0, 0) (1 0, 1, 0, 1, 0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0) (1, 0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 1, 0, 0, 0, 0, 0, 1, 0) (1 0, 0, 1, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0) (0, 1, 0, 0, 1, 0, 0, 0, 1, 0), (0, 0, 0, 1, 0, 1, 0, 0, 1, 0) (0 0, 1, 0, 0, 1, 0, 0, 1, 0, 0), (0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0) (0, 0, 1, 0, 0, 0, 1, 0, 1, 0), (0, 1, 0, 1, 0, 0, 1, 0, 1, 0) (1 0, 1, 0, 0, 0, 1, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0) (1, 0, 1, 0, 1, 0, 1, 0, 1, 0), (0, 0, 0, 1, 0, 0, 0, 0, 0, 1) (0 0, 1, 0, 0, 0, 0, 0, 0, 1, 0), (0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0) (1, 0, 0, 0, 1, 0, 0, 0, 0, 1), (0, 0, 1, 0, 1, 0, 0, 0, 0, 1) (0 1, 0, 0, 1, 0, 0, 0, 0, 1, 0), (0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0) (1, 0, 0, 0, 0, 1, 0, 0, 0, 1), (0, 1, 0, 0, 0, 0, 1, 0, 0, 1) (1 0, 0, 0, 0, 0, 1, 0, 0, 1, 0), (1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0) (1, 0, 1, 0, 0, 0, 1, 0, 0, 1), (1, 0, 0, 0, 0, 0, 0, 1, 0, 1) (1 0, 0, 1, 0, 0, 0, 1, 0, 1, 0), (0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0) (0, 1, 0, 0, 1, 0, 0, 1, 0, 1), (0, 0, 1, 0, 0, 1, 0, 1, 0, 1) (0 1, 0, 0, 0, 1, 0, 1, 0, 1, 0), (1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0) (0, 1, 0, 1, 0, 1, 0, 1, 0, 1) (0 0, 0, 0, 0, 0, 0, 0, 0, 0, 1), (0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1) (0 1, 0, 0, 0, 1, 0, 0, 0, 0, 1), (0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1) (0 1, 0, 1, 0, 0, 0, 1, 0, 0, 1), (0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1) (0 0, 1, 0, 0, 1, 0, 1, 0, 0, 1), (0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1) (0 0, 0, 1, 0, 1, 0, 0, 1, 0, 1), (0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1) (0 1, 0, 1, 0, 0, 1, 0, 1, 0, 1) Table 3: Examples of minimal dominating sets for r(10) and A(11) 2-packaging set r(ii) I A(i2) 0, 0, 0, 0, 0 0, 0, 0, 1, 0 1, 0, 1, 0, 0 1, 0, 0, 0, 1 0, 0, 1, 0, 0 0, 1, 0, 1, 0 0, 1, 0, 0, 1 0, 1, 0, 0, 0 0, 0, 0, 1, 0 1, 0, 0, 0, 1 0, 0, 1, 0, 1 0, 0, 1, 0, 0 0, 1, 0, 1, 0 0, 0, 0, 1, 0 0, 0, 1, 0, 1 0, 0, 0 1, 0, 0 0, 1, 0 0, 0, 1 0, 1, 0 0, 0, 1 1, 0, 1 1, 0, 0 1, 0, 0 0, 0, 1 0, 0, 0 0, 1, 0 0, 0, 0 0, 1, 0 1, 0, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 0, 1 0, 0, 1 0, 0, 1 1, 0, 1 0, 0, 0 0, 0, 0 1, 0, 0 1, 0, 0 0, 1, 0 0, 1, 0 (0, 0 (0, 1 (0, 0 (0, 1 (0, 0 (0, 1 (0, 0 (0, 0 (0, 1 (0, 0 (1, 0 (0, 0 (0, 0 (0, 1 (0, 1 (0, 1 (0, 0 (0, 1 (0, 0 0, 0, 0 0, 0, 1 1, 0, 0 0, 1, 0 0, 1, 0 0, 0, 0 1, 0, 1 0, 0, 1 0, 1, 0 1, 0, 0 0, 0, 0 0, 1, 0 1, 0, 1 0, 0, 0 0, 1, 0 0, 0, 0 1, 0, 0 0, 0, 1 0, 0, 0 0, 0, 0 0, 1, 0 1, 0, 1 1, 0, 0 0, 1, 0 1, 0, 0 0, 1, 0 0, 0, 1 0, 0, 0 0, 1, 0 0, 0, 1 1, 0, 1 0, 0, 0 0, 1, 0 0, 1, 0 1, 0, 1 1, 0, 0 0, 0, 0 0, 1, 0 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0 0, 1, 0 0, 0, 1 1, 0, 1 0, 0, 0 1, 0, 0 0, 1, 0 0, 1, 0 0, 0, 0 1, 0, 0 0, 1, 0 0, 0, 1 0, 0, 1 0, 0, 0 0, 1, 0 0, 0, 1 1, 0, 1 0, 0, 0 0, 1, 0 0, 0 0 0 0 ,0 Table 4: Examples of 2-packing sets for r(ii) and A(i2) A. Ilic & M. Milosevic: The parameters of Fibonacci and Lucas Cubes 29 References [1] A. Castro, S. KlavZar, M. Mollard and Y. Rho, On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes, Comput. Math. Appl. 61 (2011), 2655-2660, doi:10.1016/j.camwa.2011.03.012, http://dx.doi.org/10.1016/j.camwa.2011. 03.012. [2] P. Gregor, Recursive fault-tolerance of Fibonacci cube in hypercubes, Discrete Math. 306 (2006), 1327-1341, doi:10.1016/j.disc.2004.09.017, http://dx.doi.org/10.1016/ j.disc.2004.09.017. [3] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, volume 208 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1998. [4] W.-J. Hsu, Fibonacci cubes-a new interconnection topology, IEEE Transactions on Parallel and Distributed Systems 4 (1993), 3-12, doi:10.1109/71.205649. [5] A. Ilic, S. KlavZar and Y. Rho, Generalized Fibonacci cubes, Discrete Math. 312 (2012), 2-11, doi:10.1016/j.disc.2011.02.015, http://dx.doi.org/10.1016/j.disc.2011.02. 015. [6] S. KlavZar, On median nature and enumerative properties of Fibonacci-like cubes, Discrete Math. 299 (2005), 145-153, doi:10.1016/j.disc.2004.02.023, http://dx.doi.org/10. 1016/j.disc.2004.02.023. [7] E. Munarini, C. Perelli Cippo and N. Zagaglia Salvi, On the Lucas cubes, Fibonacci Quart. 39 (2001), 12-21. [8] E. Munarini and N. Z. Salvi, Structural and enumerative properties of the Fibonacci cubes, Discrete Math. 255 (2002), 317-324, doi:10.1016/S0012-365X(01)00407-1, Combinatorics '98 (Palermo), http://dx.doi.org/10.1016/S0012-365X(01)00407-1. [9] D. A. Pike and Y. Zou, The domination number of Fibonacci cubes, J. Combin. Math. Combin. Comput. 80 (2012), 433-444. [10] A. Taranenko and A. Vesel, Fast recognition of Fibonacci cubes, Algorithmica 49 (2007), 81-93, doi:10.1007/s00453-007-9026-5, http://dx.doi.org/10.10 07/ s00453-007-9026-5. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 31-36 A characterization of plane Gauss paragraphs Dan Archdeacon * Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA Drago Bokal, Tanja Gologranc Faculty of Natural Sciences and Mathematics, University ofMaribor, Slovenia and Institute ofMathematics, Physics and Mechanics, Ljubljana, Slovenia Received 3 May 2015, accepted 4 November 2015, published online 21 March 2016 Gauss first studied representations of self-intersecting curves in the plane using only lists of their crossings in the sequence as they occur when traversing a curve, i.e., representations using Gauss words. The characterisation of words that are Gauss words has been elusive for a long time, and only in recent decades have some good characterizations been established. Together with these, the interest in Gauss paragraphs, i.e., representations of sets of curves by sets of words listing their sequences of crossings, has came to light, and we are unaware of a (good) characterization of abstract sets of words that are Gauss paragraphs. We establish such a characterization and we show that characterizing Gauss paragraphs is algorithmically equivalent to characterizing Gauss words, as there exists a word W that can be obtained from a set of words P in linear time, such that P is a Gauss paragraph if and only if W is a Gauss word. Keywords: Gauss words, Gauss codes, Gauss paragraphs, good characterization. Math. Subj. Class.: 5C10, 57M15 1 Introduction Gauss [5,282-286] has studied representations of closed curves using lists of their crossings in the sequence obtained by following the curve. Clearly, each crossing appears exactly twice, and Gauss noticed that these two occurrences must have one an even and the other an odd index in the sequence, i.e. there has to be an odd number of letters between them. Gauss noted that the condition is not sufficient for curves with five or more crossings. The question of characterizing such words has not been solved until late 1960s, when Marx [9] and * Written posthumously. E-mail addresses: drago.bokal@um.si (Drago Bokal), tanja.gologranc@um.si (Tanja Gologranc) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 32 Ars Math. Contemp. 12 (2017) 25-29 Treybig [13] gave algorithmic characterization of words that are Gauss words. Griinbaum [6] noted that they lack the aesthetic appeal of, for instance, Kuratowski theorem, an issue resolved by Lovasz and Marx [8], who gave the first characterization satisfying Edmonds' criterion for "good characterization" [4]. Recently, the interest in Gauss words, i.e. the words that occur when the crossings of a self-intersecting curve are read in a sequence, has been renewed through several new good characterizations [3, 10, 11] and through introduction of Gauss paragraphs, sets of words corresponding in the same manner to sets of curves. The questions that arise in the bibliography are classified by Courcelle [2] into (i) Which (sets of) words over some alphabet are Gauss words (paragraphs), i.e. realizable as (sets of) (self)intersecting curves whose sequences of crossings are equal to specified (sets of) words, (ii) Which (sets of) curves can be uniquely reconstructed from their Gauss words (paragraphs) and (iii) What is the common structure of (sets of) curves having the same Gauss word (paragraph). In our paper, we investigate the question (i) for Gauss paragraphs, and develop an efficient characterization of sets of words that can be realized with sets of (self)intersecting curves in the plane so that a Gauss paragraph of this set of curves equals the original set of words. The same problem was recently studied by Schellhorn [12], who extended virtual strings introduced by Turaev [14] from single close curve S1 to sets of such curves and used them to characterize realizable Gauss paragraphs with a conjunction of seven technical conditions. In what follows, we give an elementary characterization that reduces the problem of realizability of a set of words to the problem of realizability of a single specific word obtainable from the set in linear time, avoiding the use of virtual strings. Besides showing that the problem of recognizing Gauss paragraphs is equivalent to recognizing Gauss words, the main improvement over Shcellhorn's characterization is the added algorithmic transparency. 2 Characterization of Gauss paragraphs We first summarize some of the used notation. A double-occurrence word over an alphabet E is a word in which every letter of E appears exactly twice. The double-occurrence words that are Gauss words of some self-intersecting curve have been characterized by Rosenstiehl [10,11] and de Fraysseix and de Mendez [3]. Rosenstiehl proved the following algebraic characterization of Gauss words. Theorem 2.1. [10, Theorem 2'] A double-occurrence word W on a finite set E of letters is a Gauss word if, and only if, 1. any letter of W has an even number of interlaced letters; 2. any non-interlaced pair of letters has an even number of common interlaced letters; 3. the interlaced pairs having an even number of common interlaced letters form a separating set S, i.e. there exists E' C E, such that any pair of S has a letter of E' and a letter of E \ E'. The last condition of the theorem suggests it has a natural graph-theoretic formulation. We state it in terms of the interlace-graph GW of a Gauss word W over the alphabet E, defined so that the letters of E are the vertex set, V(GW) = E, and two vertices u,v e E are adjacent in GW, uv e E(GW), if and only if they interlace in W. A cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. D. Archdeacon, D. Bokal and T. Gologranc: A characterization of plane Gauss paragraphs 33 Using these concepts, Theorem 2.1 can be stated as the following: Theorem 2.2 ([3]). Let W be a double-occurrence word over a finite alphabet E and let GW be its interlace-graph. Then W is a Gauss word if and only if 1. each component of GW is Eulerian; 2. if u and v are two nonadjacent vertices of GW, then they have an even number of common neighbors; 3. the set {e = uv | u, v have an even number of common neighbors} is a cut-set in GW. When studying sets of curves, a crossing may appear on different curves, so we need to relax the condition of double-occurrence. We define a semi-double-occurence word over an alphabet E to be a word, in which every letter of E appears at most twice. Then, a double-occurence k-paragraph1 (shortly, k-DOP) over an alphabet E is a set of k semi-double-occurence words over E, such that each letter appears precisely twice in the union of all words of the paragraph. Further, a mixed crossing of a set of (self)intersecting curves in the plane is a crossing of two different curves, i.e. not a self-crossing of some word. Correspondingly, a mixed letter of a k-DOP P is a letter that appears in two different words of P. With M (P) or just M, when the paragraph is clear from the context, we will denote the mixed letters of P. Note that, in contrast to some knot-theoretic bibliography [1], our definition follows the original definition of Gauss, which does not encode over- or under-pass information that is required for knot-theoretic investigation. For us, the curves are embedded in the plane and each crossing is either a self-crossing of some curve, appearing twice in the same word, or is a crossing of two curves, appearing once in each corresponding word. Finally, for a k-DOP P = (wi,..., wk), we define its intersection graph G(P) as the graph whose vertices are words of P, V(G(P)) = P, in which two vertices are adjacent, iff the corresponding words share a letter of E. Let P be k-DOP that contains x e M. Then we will simplify notation and write P = (xwi, XW2, . . . , Wfc ). Lemma 2.3. Let P = (xw1, xw2,..., wk) be a k-DOP and let x e M be a selected letter appearing in the first two words. Then P is a Gauss paragraph, if and only if the (k — l)-DOP Px = (xx'w1xx'w2, w3,..., wk) is a Gauss paragraph. Proof. Suppose first that P is a Gauss paragraph. Let n be a drawing that realizes P. In n, replace x in its small neighborhood by a digon xx' with incoming edges adjacent to x and outgoing to x' (see Figure 1). The resulting embedding is an embedding of (k — 1)-DOP Px, showing that Px is a Gauss paragraph. For the converse, suppose that Px is a Gauss paragraph, realized in n. We will first prove that xx' is not a cut. Indeed, if xx' is a cut, then x and x' are not interlaced in n, a contradiction. Since x and x' induce a cycle and do not induce a cut, one of the faces of this cycle is empty and the other contains the full embedding. This implies that the out-edges and the in-edges come consecutive in the vertex rotation around the empty face. By 1 As pointed out by one of the referees, a more natural name for this concept would be a sentence, as sentence is the next grammatical structure composed of words. Indeed we used double-occurrence sentence and Gauss sentence until a more thorough search through the bibliography [1] revealed that it was studied under the name Gauss paragraph. 34 Ars Math. Contemp. 12 (2017) 25-29 contracting the empty face, x and x' become a single point. By rerouting the curves so that x is a crossing, we get a realization of P. □ Figure 1: Replacing x with digon xx' or vice versa. We say that Px from Theorem 2.3 is an x-reduction of P. With a sequence of reductions, we would like to obtain a single word. Let x G w1 n w2. Since the letters appearing only in wi and w2, after x-reduction appear in a common word, at most (k — 1) reductions reduce a Gauss paragraph to a single word, to which we can apply Theorem 2.2. Let P be a k-DOP and G(P) the intersection graph of semi-double-occurence words of P; its vertices are words and two words are adjacent if they have at least one letter in common. Let T be a tree in G(P) and w1,...,wt the vertices of T, such that wj has at most one neighbor in {wi+1,..., wt} and the connecting edge results from letter m, G M. Let w1 = w1. We define recursively wl+1 = mimiwi mim'iwi+1, i = 1,... ,t — 1. The T-reduction of P is PT = (wt,wt+1,...,wfc). By induction, using the previous lemma as induction step, we get the following result: Theorem 2.4. Let P = (w1,..., wk) be a k-DOP and let T be a tree in G(P) on t vertices v1,... ,vt, such that v has at most one neighbour in {vi+1,... vt}. Then P is a Gauss paragraph, if and only if PT is a Gauss paragraph. Applying this corollary to a spanning tree of G, we get the following characterization of k-DOPs that are Gauss paragraphs: Corollary 2.5. Let P = (w1,.. .,wk). Let T be a spanning tree in G(P), and let W be the only word of PT. Then P is a Gauss paragraph, iff W is a Gauss word, i.e. iff GW satisfies the conditions of Theorem 2.2. It is clear that this corollary implies existence of a polynomial algorithm for determining whether a k-DOP is a Gauss paragraph, and thus satisfies Edmonds' criterion for a good characterization [4]: if A is the adjacency matrix of a graph G, then A2 counts the number of length-two walks between any pair of vertices, i.e., the number of common neighbors, the crucial information required for verifying conditions of Theorem 2.2. The matrix A2 can be computed in O(|V(G)|w) time, with w < 2.376, using fast matrix multiplication. This yields the dominating time-complexity term O(|£|w) of the k-DOP realizability verifying algorithm, as running time of the algorithm is dominated by the requirement to count the common neighbors of any pair of vertices of GW, either adjacent or not. There are |E| + k vertices of GW, but the vertices x and x' have the same set of neighbors and are adjacent, hence with some preprocessing it suffices to check only a matrix of size |E|. D. Archdeacon, D. Bokal and T. Gologranc: A characterization of plane Gauss paragraphs 35 Note that the best known time complexity of exact counting of all triangles in a general graph with n vertices (which is equivalent to counting the common neighbors of just the adjacent pairs of graph's vertices) is O(nu) [7], which indicates that the time complexity of checking realizability of a given k-DOW using conditions of Theorem 2.2 can hardly be improved, unless some detailed properties of the graph GW are exploited in counting the common neighbors. However, as constructing the graph G(P) can be done in time O(|E|), its spanning tree T found in O(k), and the T-reduction of P found in O(k + |E|), then any improvement in checking realizability of a double-occurring word immediately translates into an improvement of checking realizability of k-DOP. Acknowledgements This research was supported by the internationalisation of Slovene higher education within the framework of the Operational Programme for Human Resources Development 20072013 and by the Slovenian Research Agency project L7-5459. References [1] J. S. Carter, Classifying immersed curves, Proc. Amer. Math. Soc. 111 (1991), 281-287, doi: 10.2307/2047890, http://dx.doi.org/10.2307/2047890. [2] B. Courcelle, The common structure of the curves having a same Gauss word, in: Automata, universality, computation, Springer, Cham, volume 12 of Emerg. Complex. Com-put., pp. 1-37, 2015, doi:10.1007/978-3-319-09039-9, http://dx.doi.org/10.10 07/ 978-3-319-09039-9_1. [3] H. de Fraysseix and P. Ossona de Mendez, On a characterization of Gauss codes, Discrete Comput. Geom. 22 (1999), 287-295, doi:10.1007/PL00009461, http://dx.doi.org/ 10.1007/PL00009461. [4] J. Edmonds, Minimum partition of a matroid into independent subsets, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 67-72. [5] C. F. Gauss, Werke, Georg Olms Verlag, Hildesheim-New York, 1976, erganzungsreihe. IV. Briefwechsel C. F. Gauss-H. W. M. Olbers, I, Nachdruck der 1900 Auflage, herausgegeben von C. Schilling. [6] B. Grunbaum, Arrangements and spreads, American Mathematical Society Providence, R.I., 1972, conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 10. [7] M. Latapy, Main-memory triangle computations for very large (sparse (power-law)) graphs, Theoret. Comput. Sci. 407 (2008), 458-473, doi:10.1016/j.tcs.2008.07.017, http://dx. doi.org/10.1016/j.tcs.2008.07.017. [8] L. Lovasz and M. L. Marx, A forbidden substructure characterization of Gauss codes, Bull. Amer. Math. Soc. 82 (1976), 121-122. [9] M. L. Marx, The Gauss realizability problem, Proc. Amer. Math. Soc. 22 (1969), 610-613. [10] P. Rosenstiehl, A new proof of the Gauss interlace conjecture, Adv. in Appl. Math. 23 (1999), 3-13, doi:10.1006/aama.1999.0643, http://dx.doi.org/10.10 0 6/aama. 1999.0643. [11] P. Rosenstiehl and R. C. Read, On the principal edge tripartition of a graph, Ann. Discrete Math. 3 (1978), 195-226, advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). 36 Ars Math. Contemp. 12 (2017) 25-29 [12] W. Schellhorn, Virtual strings for closed curves with multiple components and filamentations for virtual links, 2015, arXiv:math/0412023. [13] L. B. Treybig, A characterization of the double point structure of the projection of a polygonal knot in regular position, Trans. Amer. Math. Soc. 130 (1968), 223-247. [14] V. Turaev, Virtual strings, Ann. Inst. Fourier (Grenoble) 54 (2004), 2455-2525 (2005), http: //aif.cedram.org/item?id=AIF_20 04_5 4_7_2 455_0. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 37-50 On skew Heyting algebras Karin Cvetko-Vah University of Ljubljana, Faculty of Mathematics and Physics, Jadranska 19, Ljubljana, Slovenia Received 29 October 2014, accepted 8 April 2016, published online 21 April 2016 Abstract In the present paper we generalize the notion of a Heyting algebra to the non-commutative setting and hence introduce what we believe to be the proper notion of the implication in skew lattices. We list several examples of skew Heyting algebras, including Heyting algebras, dual skew Boolean algebras, conormal skew chains and algebras of partial maps with poset domains. Keywords: Skew lattices, Heyting algebras, non-commutative algebra, intuitionistic logic. Math. Subj. Class.: 06F35, 03G27 1 Introduction Non-commutative generalizations of lattices were introduced by Jordan [11] in 1949. The current approach to such objects began with Leech's 1989 paper on skew lattices [13]. Similarly, skew Boolean algebras are non-commutative generalizations of Boolean algebras. In 1936 Stone proved that each Boolean algebra can be embedded into a field of sets [20]. Likewise, Leech showed in [14, 15] that each right-handed skew Boolean algebra can be embedded into a generic skew Boolean algebra of partial functions from a given set to the codomain {0,1}. Bignall and Leech [5] showed that skew Boolean algebras play a central role in the study of discriminator varieties. Though not using categorical language, Stone essentially proved in [20] that the category of Boolean algebras and homomorphisms is dual to the category of Boolean topo-logical spaces and continuous maps. Generalizations of this result within the commutative setting yield Priestley duality [16, 17] between bounded distributive lattices and Priestley spaces, and Esakia duality [9] between Heyting algebras and Esakia spaces. (See [4] for details.) In a recent paper [10] on Esakia's work, Gehrke showed that Heyting algebras may be understood as those distributive lattices for which the embedding into their Booleani-sation has a right adjoint. A recent line of research generalized the results of Stone and E-mail address: karin.cvetko@fmf.uni-lj.si (Karin Cvetko-Vah) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 38 Ars Math. Contemp. 12 (2017) 25-29 Priestley to the non-commutative setting. By results in [1] and [12], any skew Boolean algebra is dual to a sheaf of rectangular bands over a locally-compact Boolean space. A further generalization given in [2] showed that any strongly distributive skew lattice (as defined below) is dual to a sheaf (of rectangular bands) over a locally compact Priestley space. While Boolean algebras provide algebraic models of classical logic, Heyting algebras provide algebraic models of intuitionistic logic. In the present paper we introduce the notion of a skew Heyting algebra. In passing to the non-commutative setting one needs to sacrifice either the top or the bottom of the algebra in order not to end up in the commutative setting. In the previous papers [1], [12] and [2] algebras with bottoms were considered, and hence the notion of distributivity was generalized to the notion of so-called strong distribu-tivity. If one tried to define an implication operation in the setting of strongly distributive skew lattices with a bottom as a right adjoint to conjunction, that would force the skew lattice to also possess a top and hence be commutative, resulting in a usual Heyting algebra. In order to define implication in the skew lattice setting we consider the V - A duals of strongly distributive skew lattices with a bottom, namely, the co-strongly distributive skew lattices with a top. That is not surprising as a top plays a crucial role in logic. The category of co-strongly distributive skew lattices with a top is, of course, isomorphic to the category of strongly distributive skew lattices with a bottom. In choosing co-strongly distributive skew lattices with a top we follow the path paved by Bignall and Spinks in [6], and by Spinks and Veroff in [19] where dual skew Boolean algebras were introduced. For further reading on implications in skew Boolean algebras and their algebraic duals, see [7]. After reviewing some preliminary definitions and concepts in Section 2, in the next section we introduce the notion of a skew Heyting algebra, prove that such algebras form a variety and show that the maximal lattice image of a skew Heyting algebra is a generalized Heyting algebra (possibly without a bottom). Indeed, a co-strongly distributive skew lattice with a top is the reduct of a skew Heyting algebra, if and only if its maximal lattice image forms a generalized Heyting algebra. (See Theorem 3.5.) This leads to a number of useful corollaries and examples. We finish with Section 4 where we explore the consequences of our results to duality theory, and describe how skew Heyting algebras correspond to sheaves over local Esakia spaces. 2 Preliminaries A skew lattice is an algebra S = (S; A, V) of type (2,2) such that A and V are both idempotent and associative and they satisfy the following absorption laws: x A (x V y) = x = x V (x A y) and (x A y) V y = y = (x V y) A y. These identities are collectively equivalent to the pair of equivalences: x Ay = x ^ xVy = y and x A y = y ^ x V y = x. On a skew lattice S one can define the natural partial order by stating that x < y if and only if x V y = y = y V x, or equivalentely x A y = z = y A x, and the natural preorder by x ^ y if and only if y V x V y = y, or equivalentely x A y A x = x. Green's equivalence relation D is then defined by xDy if and only if x ^ y and y ^ x. (2.1) K. Cvetko-Vah: On skew Heyting algebras 39 Lemma 2.1. ([8]). For elements x and y elements of a skew lattice S the following are equivalent: (i) x < y, (ii) x V y V x = y, (iii) y A x A y = x. Leech's First Decomposition Theorem for skew lattices states that the relation D is a congruence on a skew lattice S, S/D is the maximal lattice image of S, and each congruence class is a maximal rectangular skew lattice in S [13]. Rectangular skew lattices are characterized by x A y A z = x A z, or equivalentely x V y V z = x V z. We denote the D-class containing an element x by Dx. It was proved in [13] that a skew lattice always forms a regular band for either of the operations A, V, i.e. it satisfies the identities x A u A x A v A x = x A u A v A x and x V u V x V v V x = x V u V v V x. A skew lattice with top is an algebra (S; A, V, 1) of type (2, 2,0) such that (S; A, V) is a skew lattice and x V 1 = 1 = 1 V x, or equivalently x A 1 = x =1 A x, holds for all x g S. A skew lattice with bottom is defined dually and the bottom, if it exists, is usually denoted by 0. Furthermore, a skew lattice is called strongly distributive if it satisfies the following identities: x A (y V z) = (x A y) V (x A z) and (x V y) A z = (x A z) V (y A z); and it is called co-strongly distributive if it satisfies the identities: x V (y A z) = (x V y) A (x V z) and (x A y) V z = (x V z) A (y V z). If a skew lattice S is either strongly distributive or co-strongly distributive then S is distributive in that it satisfies the identities x A (y V z) A x = (x A y A x) V (x A z A x) and x V (y A z) V x = (x V y V x) A (x V z V x). A skew lattice S that is jointly strongly distributive and co-strongly distributive is binormal, i.e. S factors as a direct product of a lattice L and a rectangular skew lattice B, S = L x B, with L in this case being distributive. (See [15] and [18].) Applying duality to aresult of Leech [15], it follows that a skew lattice S is co-strongly distributive if and only if S is jointly: • quasi-distributive: the maximal lattice image S/D is a distributive lattice, • symmetric: x A y = y A x if and only if x V y = y V x, and • conormal: x V y V z V w = x V z V y V w. If a skew lattice is conormal then given any u g S the set ut = {u V x V u | x G S} = {x G S | u < x} forms a (commutative) lattice for the induced operations A and V, cf. [15]. The following lemma is the dual of a well known result in skew lattice theory. 40 Ars Math. Contemp. 12 (2017) 25-29 Lemma 2.2. Let S be a conormal skew lattice and let A and B be D-classes such that B < A holds in the lattice S/D. Given b G B there exists a unique a G A such that b < a. Proof. First the uniqueness. If a and a' both satisfy the desired property then by Lemma 2.1 we have a = b V a V b and likewise a' = b V a' V b. Now, using idempotency of V, conormality and the fact that a D a' we obtain: a = b V a V b = b V a V a' V a V b = b V a V a' V b = b V a' V a V a' V b = b V a' V b = a'. To prove the existence of a take any x G A and set a = b V x V b. Then a G A and using the idempotency of V we get: b V a V b = b V (b V x V b) V b = b V x V b = a which implies b < a. □ An important class of strongly distributive skew lattices that have a bottom is the class of skew Boolean algebras where by a skew Boolean algebra we mean an algebra S = (S; A, V, \, 0) where (S; A, V, 0) is a strongly distributive skew lattice with bottom 0, and \ is a binary operation on S such that both (x A y A x) V (x \ y) = x = (x \ y) V (x A y A x) and (x A y A x) A (x \ y) = 0 = (x \ y) A (x A y A x). Given any element u of a skew Boolean algebra S the set u^ = {u A x A u | x G S} = {x G S | x < u} is a Boolean algebra with top u and with u \ x being the complement of u A x A u in u^. Recall that a Heyting algebra is an algebra H = (H; A, V, 1,0) such that (H, A, V, 1,0) is a bounded distributive lattice that satisfies the condition: (HA) x A y < z iff x < y ^ z. Stated otherwise, Vy, z G H the sublattice {x G H | x A y < z} is nonempty and contains a top element to be denoted by y ^ z. Thus, given a bounded distributive lattice (H; A, V, 1,0), if a binary operation ^ exists that makes (H; A, V, 1,0) a Heyting algebra, then it is unique because it is already there implicitly. Indeed, given two isomorphic lattices, if either is the lattice reduct of a Heyting algebra then so is the other, and both are isomorphic as Heyting algebras. Equivalently, (HA) can be replaced by the following set of identities: (H1) (x ^ x) = 1, (H2) x A (x ^ y) = x A y, (H3) y A (x ^ y) = y, (H4) x ^ (y A z) = (x ^ y) A (x ^ z). Lemma 2.3. In any Heyting algebra, x ^ y = (x V y) ^ y. A generalized Heyting algebra is an algebra A = (A; A, V, 1) such that the reduct (A, A, V, 1) is a distributive lattice with top 1, and condition (HA) holds. If it also has a bottom, it is a Heyting algebra. In general, each upset ut forms a Heyting algebra. The identities above also characterize this more general class of algebras, which are often called Brouwerian algebras. K. Cvetko-Vah: On skew Heyting algebras 41 3 Skew Heyting algebras A skew Heyting lattice is an algebra S = (S; A, V, 1) of type (2,2,0) such that: • (S; A, V, 1) is a co-strongly distributive skew lattice with top 1. Each upset u sup is thus a bounded distributive lattice. • for any u g S an operation can be defined on ut such that (ut; A, V, 1, u) is a Heyting algebra with top 1 and bottom u. Given a skew Heyting lattice S, define ^ on S by setting x ^ y = (y V x V y) ^y y. A skew Heyting algebra is an algebra S = (S; A, V, 1) of type (2, 2, 2,0) such that (S; A, V, 1) is a skew Heyting lattice and ^ is the implication thus induced. A sense of global coherence for ^ on S is given by: Lemma 3.1. Let S be a skew Heyting lattice with ^ as defined above and let x, y, u g S be such that both x, y G ut hold. Then x ^ y = x y. Proof. As x and y both lie in ut, they commute. By the definition of x ^ y = (x V y) y > y by (H3). On the other hand, since is the Heyting implication in the Heyting algebra ut it follows that x y = (x V y) y > y. Thus y, x V y, (x V y) y and (x V y) y all lie iin the Heyting algebra yt. The maximal element characterization of both (x V y) y and (x V y) y forces both to agree. □ We will use the axioms of Heyting algebras to derive an axiomatization of skew Heyting algebras. The reader should find most of the axioms of Theorem 3.2 below to be intuitively clear generalizations to the non-commutative case. However, two axioms should be given further explanation. Firstly, the u in axiom (SH4) below appears since upon passing to the non-commutative case, an element that is both below x and y with respect to the partial order < no longer need exist. (We can have x A y A x < x but not x A y A x < y in general.) Similarly, axiom (SH0) is needed since in the non-commutative case it no longer follows from the other axioms, the reason being that in general x < y V x V y need not hold. Theorem 3.2. Let (S; A, V, 1) be an algebra of type (2,2, 2,0) such that (S; A, V, 1) is a co-strongly distributive skew lattice with top 1. Then (S; A, V, 1) is a skew Heyting algebra if and only if it satisfies the following axioms: (SH0) x ^ y = (y V x V y) ^ y. (SH1) x ^ x =1, (SH2) x A (x ^ y) A x = x A y A x, (SH3) y A (x ^ y) = y and (x ^ y) A y = y, (SH4) x ^ (u V (y A z) V u) = (x ^ (u V y V u)) A (x ^ (u V z V u)). Proof. Assume that S is a skew Heyting algebra. (SH0). Both x ^ y and (y V x V y) ^ y are defined as (y V x V y) y. Thus they are equal. (SH1). This is true because 1 A x = x is true in xt. 42 Ars Math. Contemp. 12 (2017) 25-29 (SH2). In yt (H2) implies (y V x V y) A ((y V x V y) ^y y) = (y V x V y) A y = y. Hence x A (y V x V y) A (x ^ y) A x = x A y A x. On the other hand, x A (y V x V y) A (x ^ y) A x = x A (y V x V y) A x A (x ^ y) A x = x A (x ^ y) A x, where we have used the regularity of A and the fact that x ^ y V x V y. (SH3). Both identities hold because y A (y V x V y) = y in yt. Thus x ^ y = (y V x V y) ^ y > y. (SH4). First note that (SH4) is equivalent to (SH4') (u Vx Vu) ^ (u V (y A z) Vu) = ((u Vx Vu) ^ (u Vy Vu)) A ((u Vx Vu) ^ (u V z V u)). Indeed, (SH0) and the conormality of V give both (u V x V u) ^ (u V w V u) = (u V x V w V u) ^ (u V w V u) and x ^ (u V w V u) = (u V x V w V u) ^ (u V w V u) so that x ^ (u V w V u) = (u V x V u) ^ (u V w V u). Hence it suffices to prove that (SH4') holds. Observe that distributivity implies (u V y V u) A (u V z V u) = u V (y A z) V u. (3.1) Since u V x V u, u V y V u, u V z V u and u V (y A z) V u all lie in ut we can simply compute in ut. Using (3.1) and axiom (H4) for Heyting algebras we obtain: (u V x V u) ^ (u V (y A z) V u) = (u V x V u) ^ ((u V y V u) A (u V z V u)) = ((u V x V u) ^ (u V y V u)) A ((u V x V u) ^ (u V z V u)). To prove the converse assume that (SH0)-(SH4) hold. Given arbitrary u G S and x, y, z G ut set x y = x ^ y. Axiom (SH3) implies that x ^ y G yt C ut. Thus the restriction of ^ to ut is well defined. Since ut is commutative with bottom u, axioms (SH1)-(SH4) for ^ respectively simplify to (H1)-(H4) for making the Heyting implication on ut. Axiom (SH0) assures that ^ is indeed the skew Heyting implication satisfying a ^ b = (b V a V 6) ^b b, for any a, 6 G S. □ Corollary 3.3. Skew Heyting algebras form a variety. In the remainder of the paper, given a skew Heyting algebra we shall simplify the notation to ^ when referring to the Heyting implication in ut. Remarks made about Heyting algebras in Section 2 apply here also. Given a co-strongly distributive skew lattice (S; A, V, 1) with a top 1, if a binary operation ^ exists that makes (S; A, V, 1) a skew Heyting algebra, then it is unique since it is already there implicitly. Hence, given two isomorphic skew lattices, if either is the reduct of a skew Heyting algebra, then so is the other and both are isomorphic as skew Heyting algebras. Proposition 3.4. The relation D defined in (2.1) is a congruence on any skew Heyting algebra. K. Cvetko-Vah: On skew Heyting algebras 43 Proof. Let (S; A, V, 1) be a skew Heyting algebra. Since D is a congruence for co-strongly distributive skew lattices with a top, we only need to prove (a ^ b) D (c ^ d) holds for any a, b, c, d G S satisfying a D c and b D d. Without loss of generality we may assume b < a and d < c. (Otherwise replace a by b V a V b and c by d V c V d.) To begin, define a map y : bf ^ df by setting y(x) = d V x V d. We claim that y is a lattice isomorphism of (bf; A, V) with (df; A, V), with inverse ^ : df ^ bf given by ^(y) = b V y V b. It is easily seen that y and ^ are inverses of each other. For instance, ^(y(x)) = b V d V x V d V b equals (b V d V b) V x V (b V d V b) by the regularity of V. But the latter reduces to b V x V b because bDd, and since x G bf it reduces further to x by Lemma 2.1, giving ^(y(x)) = x. y must preserve A and V. Indeed distributivity gives: y(x A x') = d V (x A x') V d = (d V x V d) A (d V x' V d) = y(x) A y(x'). And the regularity gives: y(x V x') = d V (x V x') V d = (d V x V d) V (d V x' V d) = y(x) V y(x'). Thus y (and is a lattice isomorphism of bf with df. But then y and ^ are also isomorphisms of Heyting algebras. That is, e.g., y(x ^ y) = y(x) ^ y(y). Next, observe that x D y(x) for all x G bf. Indeed, regularity gives: y(x) V x V y(x) = (d V x V d) V x V (d V x V d) = d V x V d = y(x) and likewise x V y(x) V x = ^(y(x)) V y(x) V ^(y(x)) = ^(y(x)) = x. There is more: a is the unique element in its D-class belonging to bf and c is the unique element in the same D-class belonging to df (since each upset uf intersects any D-class in at most one element). But y(a) in df behaves in the manner just like c, and so y(a) = c. Since b D d, y(b) = d V b V d = d and y(a ^ b) = y(a) ^ y(b) = c ^ d, thus giving a ^ b D c ^ d. □ Following [5] a commutative subset of a symmetric skew lattice is a non-empty subset whose elements both join and meet commute. Theorem 3.5. Given a co-strongly distributive skew lattice (S; A, V, 1) with top 1, an operation ^ can be defined on S making (S; A, V, ^, 1) a skew Heyting algebra if and only if the operation ^ can be defined on S/D making (S/D; A, V, Di) a generalized Heyting algebra. Proof. To begin, for any u in S, the upset uf is a D-class transversal of the principal filter S V u V S. Secondly, the induced homomorphism y : S ^ S/D is bijective on any commutative subset of S since distinct commuting elements of S lie in distinct D-classes. It follows that for each u in S, y restricts to an isomorphism of upsets, yu : uf = y(u)f. Thus each upset uf in S forms a Heyting algebra if and only if each upset vf in S/D, being some y(u)f, must form a Heyting algebra. The theorem follows. □ Comment. This result is a near-dual of the important fact that a strongly distributive skew lattice S with bottom 0 is the (necessarily unique) reduct of a skew Boolean algebra if and only if its lattice image S/D is the reduct of a (necessarily unique) generalized Boolean algebra. ([15], 3.8.) 44 Ars Math. Contemp. 12 (2017) 25-29 We next consider consequences of the above theorem. The first is on the "skew lattice side" of things and the next is more on the "Heyting side". But first recall the definitions of Green's relations L and R on a skew lattice: xLy ^ (x A y = x & y A x = y, or equivalently x V y = y & y V x = x), xRy ^ (x A y = y & y A x = x, or equivalently x V y = x & y V x = y). Relations L and R are contained in the Green's relation D defined above and L o R = R o L = D holds. A skew lattice is called right-handed if the relation L is trivial, in which case D = R, and it is called left-handed if the relation R is trivial, in which case D = L. By Leech's Second Decomposition Theorem [13] the relations L and R are congruences on any skew lattice S, S/R is left-handed, S/L is right-handed and the following diagram is a pullback: S-S/R S/L-S/D Corollary 3.6. If S = (S; A, V, 1) be a co-strongly distributive skew lattice with top 1, then the following are equivalent: 1. S is the reduct of a skew Heyting algebra (S; A, V, 1). 2. S/L is the reduct of a skew Heyting algebra (S/L; A, V, 1). 3. S/R is the reduct of a skew Heyting algebra (S/R; A, V, 1). Moreover, both L and R are congruences on any skew Heyting algebra. Proof. The equivalence of (i)-(iii) is due to the preceding theorem plus the fact that S/D, (S/L)/Ds/L and (S/R)/Ds/R are isomorphic. Next, the induced map p : S ^ S/L is at least a homomorphism of co-strongly distributive skew lattices. By the argument of the preceding theorem, it induces isomorphisms between corresponding pairs of upsets, ut in S and L„t in S/L. Thus given x ^ y = (y V x V y) ^y y and u ^ v = (v V u V v) ^v v with x;, L;, u and y;, L;, v in S, both (y V x V y) y and (v V u V v) v are mapped to the common LyVxvy Ly, making x ^ y;, L;, y ^ v in S. A similar argument applies to the induced map A : S ^ S/R. □ An alternative to the characterization of Theorem 3.2 is given by: Corollary 3.7. Every skew Heyting algebra satisfies: (SHA) x ^ y ^ z if and only if x A y ^ z. In particular, x ^ y =1 iff x ^ y. In general, an algebra S = (S; A, V, 1) of type (2,2, 2,0) is a skew Heyting algebra if the following conditions hold: 1. The reduct (S; A, V, 1) is a co-strongly distributive skew lattice with top 1. 2. y < x ^ y holds for all x, y G S. K. Cvetko-Vah: On skew Heyting algebras 45 3. S satisfies axiom (SHA). Proof. Given that S is a skew Heyting algebra, since the induced epimorphism ^ : S ^ S/D is a homomorphism of skew Heyting algebras we have x ^ y ^ z iff y(x) < y(y) ^ y(z) iff y(x) A y(y) < y(z) iff x A y ^ z. Conversely, let S = (S; A, V, 1) be an algebra of type (2, 2,2,0) satisfying (1)-(3). Suppose that x, y, z lie in a common upset u t. Since < is just < in u t nad y ^ z lies in u t by (2) we have x < y ^ z iff x A y < z in u t. (S; A, V, 1) is thus at least a skew Heyting lattice. Now consider the derived implication given by x y = (y V x V y) ^y y. Both y ^ z and y z satisfy (SHA) and thus are D-equivalent. But since both lie in the sublattice zt, they must be equal. □ We have seen that each skew Heyting algebra is basically a co-strongly distributive skew lattice S with top, say 1, for which S/D is a generalized Heyting algebra, in which case the Heyting structure on each upset ut of S is obtained from that of the isomorphic upset D„t in S/D. This suggests that all standard classes of generalized Heyting algebras yield classes of skew Heyting algebras whose maximal commutative images belong to the particular class. We consider several cases. Case 1. Finite distributive lattices possess a well-defined Heyting algebra structure. Thus any finite co-strongly distributive skew lattice with a top, or more generally any co-strongly distributive skew lattice with a top and a finite maximal lattice image is the reduct of a unique skew Heyting algebra. Case 2. All chains possessing a top 1 form Heyting algebras. Here things are simple: Thus all co-strongly distributive skew chains with a top are skew Heyting algebra reducts, where a skew chain is any skew lattice S where S/D is a chain, i.e., ^ is a total preorder on S. Here, given x, y in a common ut one has: Case 3. Dual generalized Boolean algebras. These are algebras S = (S; A, V, \\, 1) where (S; A, V, 1) is a distributive lattice with top 1 and \\ is a binary operation on S such that (y V x) V (y \\ x) = 1 and (y V x) A (y \\ x) = y for all x, y in S. As with \ for generalized Boolean algebras, \\ is uniquely determined. Moreover, in this case, x ^ y = y \\ x. A dual-skew Boolean algebra S = (S; A, V, \\, 1) is an algebra such that (S; A, V, 1) is a co-strongly distributive skew lattice with top 1 and binary operation \\ such that: x ^ y 1; if x < y. y; otherwise. x ^ y 1; if x ^ y. y; otherwise. (y V x V y) V (y \\ x) = 1 = (y \\ x) V (y V x V y); (y V x V y) A (y \\ x) = y = (y \\ x) A (y V x V y). 46 Ars Math. Contemp. 12 (2017) 25-29 The relevant diagram is: 1 y V x V y y \\ x y These dual algebras are, of course, precisely the co-strongly distributive skew lattices with a top whose maximal lattice images are the lattice reducts of dual-generalized Boolean algebras. Once again we follow the commutative case: x ^ y = y \\ x which now is y \\ (y V x V y) in yt We thus have: Corollary 3.8. A co-strongly distributive skew lattice with a top S = (S; A, V, 1) is the reduct of a uniquely determined skew Heyting algebra (S; A, V, \\, 1) if any one of the following conditions holds: 1. S/D is finite. 2. S is a skew chain. 3. S is the reduct of a dual generalized Boolean algebra, S = (S; A, V, \\, 1). Implicit in Case 3 is a basic duality that occurs for skew lattices. Given a skew lattice S = (S; A, V), its (vertical) dual is the skew lattice S^ = (S; V$), where as binary functions, A^ = V and V^ = A. Clearly S^ = S and any homomorphism f : S ^ T of skew lattices ia also a homomorphism from S^ to T$; moreover a skew lattice S is distributive (or symmetric) iff S^ is thus. Either S or S^ is strongly distributive iff the other is co-strongly distributive; more generally, S or S^ is normal iff the other is co-normal. Also, one has a bottom element iff the other has a top element, both being the same element in S. Expanding the signature, (S; A, V, \, 0) is a skew Boolean algebra if and only if its dual (S; A$, V$, \\, 1) is a dual skew Boolean algebra where \ and 0 are replaced by \\ and 1. Thus any skew Boolean algebra (S; A, V, \, 0) induces a skew Heyting algebra (S; A$, V$, 1) where x ^ y = y \ x and 1 = old 0. Standard examples of skew Boolean algebras thus give us: Example 3.9. Given sets X and Y, the skew Heyting operations derived from the skew Boolean operations on the set P(X, Y) of all partial functions from X to Y are as follows. skew Heyting operation description skew Boolean operation f A g f U (g|domg-domf ) f V g f V g g|domgndom/ f A g f ^ g g|domg- domf g \ f 1 0 0 K. Cvetko-Vah: On skew Heyting algebras 47 Example 3.10. Given a surjective function n : Y ^ X, let Sec(n) denote the set of all sections of n, that is, functions f from subsets U of X to Y such that n o f = iddom(f ). Skew Heyting algebra operations and corresponding skew Boolean operations are defined on Sec(n) using precisely the above descriptions. At first glance this seems to be just a subalgebra of the type of algebra in Example 1. The latter, however, is isomorphic to Sec(n) where n is now the coordinate projection of X x Y onto X. Modifications of this example arise in the dualities of the next section. It so happens that any right-handed (co-)strongly distributive skew lattice is isomorphic to a subset of partial functions in some P (X, Y ) that is closed under the relevant A and V operations above. (See [12] Section 3.7.) It follows that the skew lattice reduct of a skew Heyting algebra is isomorphic to some such partial function algebra. The difference of this more general case from that of the two examples above is that here x ^ y need not have a polynomial definition, unlike the co-Boolean case where x ^ y = y \ x. The following result is useful for computing in skew Heyting algebras. Proposition 3.11. Let S = (S; A, V, 1) be a skew Heyting algebra and x, y, z G S. Then (x V y V x) ^ z = (x ^ z) A (y ^ z) A (x ^ z). Proof. As S/D is a generalized Heyting algebra and relation D respects all skew Heyting algebra operations, it follows that (x V y V x) ^ z D (x ^ z) A (y ^ z) A (x ^ z). However, both (x V y V x) ^ z and (x ^ z) A (y ^ z) A (x ^ z) are above z with respect to the natural partial order, and hence must be equal by Lemma 2.2. □ A skew lattice S is meet [join] complete if each nonempty commutative subset possesses an infimum [a supremum] in S. It follows from the dual of [5] Proposition 2.10 that if S is a meet complete co-strongly distributive skew lattice with 1, then S is complete. We call a skew Heyting algebra complete if it is complete as a skew lattice. 4 Connections to duality Dual skew Boolean algebras are order duals (upside-downs) to usually studied skew Boolean algebras. Skew Boolean algebras and dual skew Boolean algebras are categorically isomorphic. Right-handed (dual) skew Boolean algebras are dually equivalent to sheaves over locally compact Boolean spaces by results of [1] and [12], where a locally compact Boolean space is a topological space whose one-point-compactification is a Boolean space. The obtained duality asserts that any right- [left-]handed skew Boolean algebra is isomorphic to the skew Boolean algebra of compact open sections (i.e. sections over compact open subsets) of an etale map over some locally compact Boolean space. Let us note that the restriction to right- [left-]handed algebras is not a major restriction since Leech's Second Decomposition Theorem yields that any skew lattice is a pull back of a left-handed and a right-handed skew lattice over their common maximal lattice image [13]. The general two-sided case was also covered in [1]. Bounded distributive lattices are dual to Priestley spaces; in this duality each bounded distributive lattice is represented as the distributive lattice of all clopen (i.e. closed and open) upsets of a Priestley space. The Esakia duality established in [9] yields that Heyting algebras are dual to Esakia spaces, i.e. those Priestley spaces in which the downset of each 48 Ars Math. Contemp. 12 (2017) 25-29 clopen set is again clopen. Moreover, if (X, <, t) is an Esakia space then given clopen subsets U and V in X the implication is defined by U ^ V = X \|(U \ V). Duality for strongly distributive skew lattices was recently established in [2]. It yields that right-handed strongly distributive skew lattices with bottom are dual to the sheaves over locally Priestley spaces, where by a locally Priestley space we mean an ordered topological space whose one-point-compactification is a Priestley space. Via the obtained duality each right-handed strongly distributive skew lattice with bottom is represented as a skew lattice of sections over copen (i.e. compact and open) downsets of a locally Priestley space, with the operations being defined as follows: 0 = 0, r ^ S = s|domrndoms, r V S = r U s|doms-domr, r \ s ^ r \ |domr-doms * Given a distributive lattice L denote by Lc the distributive lattice that is obtained from L by reversing the order. Denote by DL the category of all distributive lattices, by PS the category of all locally Priestley spaces and consider the functors: c : DL ^ DL r : PS ^ PS L ^ Lc and (X, <) ^ (X, >). Restricting the functorsc and r to the categories HA of all Heyting algebras and ES of all Esakia spaces, respectively, yields the following isomorphism of categories: c : HA ^ cHA , r : ES ^ cES L ^ Lc and (X, <) ^ (X, >), where cHA denotes the category of all co-Heyting algebras (defined as order-inverses of Heyting algebras) and cES denotes the category of all co-Esakia spaces the latter being introduced in [3] as Priestley spaces in which an upset of a clopen is again clopen. We introduce the following categories: SDSL : strongly distributive skew lattices with 0, cSDSL : co-strongly distributive skew lattices with 1, SHA : skew Heyting algebras, cSHA : co-skew Heyting algebras, with the latter being defined as the category of all algebras of the form Sc, where S is a skew Heyting algebra and c : SDSL ^ cSDSL S ^ Sc is the isomorphism of categories that turns a skew lattice upside-down. The restriction of the functorc to the categories cSHA and SHA yields the isomorphism: c : cSHA ^ SHA S ^ Sc . K. Cvetko-Vah: On skew Heyting algebras The isomorphism of categories induces an isomorphism of concepts: 49 SHA cSHA A V 1 strong codistributivity filter prime filter V A 0 strong distributivity ideal prime ideal It follows from Theorem 3.5 that the skew Heyting algebra structure can be imposed exactly on those co-strongly distributive skew lattices with top whose maximal lattice image is a generalized Heyting algebra. Therefore the duality for right-handed skew Heyting algebras yields that they are dual to sheaves over local Esakia spaces, i.e. ordered topological spaces whose one-point-compactification is an Esakia space. Let (B, <) be an Esakia space, E a topological space and p : E ^ B a surjective etale map. Consider the set S of all sections of p over copen upsets in B, i.e. an element of S is a map s : U ^ E, where U is a copen upset in B, that satisfies the property p o s = idy. A section s G S is considered to be below a section r G S when s extends r. The skew Heyting operations are defined on S by: r V s = s 1 domrHdoms , r A s = r U s|doms\domr , r ^ s = r||(doms\domr) 1 = 0. Theorem 4.1. Let p : E ^ B be a surjective etale map over a local Esakia space B. Then the set S of all sections of p over copen upsets in B forms a skew Heyting algebra under the above operations. References [1] A. Bauer and K. Cvetko-Vah, Stone duality for skew Boolean algebras with intersections, Houston J. Math. 39 (2013), 73-109. [2] A. Bauer, K. Cvetko-Vah, M. Gehrke, S. van Gool and G. Kudryavtseva, A Non-Commutative Priestley Duality, Topology andAppl. 160 (2013), 1423-1438. [3] G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia and A. Kurz, Bitopological duality for distributive lattices and Heyting algebras, Math. Structures Comput. Sci. 20 (2010), 359-393. [4] N. Bezhanishvili, Lattices of Intermediate and Cylindric Modal Logics, PhD Thesis, ILLC, University of Amsterdam, 2006. [5] R. J. Bignall and J. E. Leech, Skew Boolean algebras and discriminator varieties, Alg. Universalis 33 (1995), 387-398. [6] R. J. Bignall and M. Spinks, Some New Strategies for Theorem-Proving in First Order Logic with Equality, Proc. ISMVL 1997/98. [7] R. J. Bignall, M. Spinks and R. Veroff, Discriminator logics (Research announcment), Australasian Journal of Logic (11:2) (2014), 159-171. 50 Ars Math. Contemp. 12 (2017) 25-29 [8] J. P. Pita da Costa, On the Coset Structure of a Skew Lattice, Demonstratio Mathematica XLIV No. 4(2011), 673-692. [9] L. L. Esakia, Topological Kripke models, Soviet Mathematics Doklady 15 (1974), 147-151. [10] M. Gehrke, Canonical extensions, Esakia spaces, and universal models, Chapter in Leo Esakia on duality in modal and intuitionistic logics, upcoming special volume in the series Trends in Logic (Outstanding Contributions subseries) dedicated to the achievements of Leo Esakia, 2014. [11] P. Jordan, Uber nichtkommutative Verbande, Arch. Math. 2 (1949), 56-59. [12] G. Kudryavtseva, A refinement of Stone duality to skew Boolean algebras, Algebra Universalis 67 (2012), 397-416. [13] J. Leech, Skew lattices in rings, Alg. Universalis 26 (1989), 48-72. [14] J. Leech, Skew Boolean Algebras, Alg. Universalis 27 (1990), 497-506. [15] J. Leech, Normal skew lattices, Semigroup Forum 44 (1992), 1-8. [16] H. A. Priestley, Representation of distributive lattices by means of ordered stone spaces, Bull. Lond. Math Soc. 2 (1970), 186-190. [17] H. A. Priestley, Ordered topological spaces and the representation of distributive lattices, Proc. Lond. Math Soc. (3) 24 (1972), 507-530. [18] B. M. Schein, Pseusosemilattices and Pseudolattices, Amer. Math. Soc. Transl. (2) 119 (1983), 1-16. [19] M. Spinks and R. Veroff, Axiomatizing the skew Boolean propositional calculus, J. Automat. Reason. 37 (2006), no. 1-2, 3-20. [20] M. H. Stone, The theory of representation for Boolean algebras, Trans. Amer. Math. Soc. 74 (1936), 37-111. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 51-65 A decomposition for Markov processes at an independent exponential time Received 29 September 2015, accepted 12 October 2015, published online 22 April 2016 The path of Markov process X run up to an independent exponential random time Sg can be decomposed into the part prior to the last exit time from a point before Sg, and the remainder up to Sg. In this paper the laws of the two segments are identified under suitable assumptions using excursion theory. Keywords: Markov processes, excursions, last exit decomposition, diffusions, Brownian motion. Math. Subj. Class.: 60J25, 60J55, 60J60, 60J65, 60G51. 1 Introduction Considering a Markov process X up to an independent exponential time Sg with rate 6 > 0 has been used effectively to compute functionals of X. The computations are often based on decompositions of the path of X up to Sg into fragments before and after the last exit time from a set before time Sg. The known results described below are more general in the sense that the path is decomposed at the last exit from a set before either fixed times or random times belonging to a suitable family. Choosing an independent exponential time in some cases leads to simpler descriptions of the laws of the two fragments involved. They are often conditionally independent given suitable conditioning variables and their laws are related to laws of known processes. Williams [27] uses a decomposition of Brownian motion with drift run up to an independent Sg to prove result of Ray [23] on the distribution of local times in the space *This work is supported in part by the Slovenian Research Agency (research program P1-0285). E-mail address: mihael.perman@fmf.uni-lj.si (Mihael Perman) Mihael Perman * Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia and Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 52 Ars Math. Contemp. 12 (2017) 25-29 variable. In their investigations of Ray-Knight theorems for Brownian motion B at fixed times Biane and Yor [4] considered the pair of processes (Bt: 0 < gSe) and (BSg -t: 0 < t < Sg — gSg) where Sg is an exponential random variable independent of B and gt — sup{s < t: Bs =0} is the last exit time from 0 before time t. Under P0 the two processes are shown to be independent and their conditional laws given the local time L(Sg) at zero of Brownian motion and BSg respectively are identified. This decomposition has been exploited by Jeanblanc, Pitman and Yor [13] to derive Feynman-Kac formulae for Brownian motion. Salminen, Vallois and Yor [26] extend the decomposition for Brownian motion to linear diffusions on [0, to) with 0 a recurrent point and use them to study the excursion of the diffusion straddling an independent exponential time. For general Markov processes Pittenger and Shih [22] investigated the dependence of the fragments of the path of a cadlag strong Markov process X before coterminal time Lt < t and the fragment on the interval between Lt and t. Last exit times LfF before time t from a closed set F are coterminal times. It is shown that given a suitably defined a-algebra FLt, the conditional law of the process (XLt+s: 0 < s < t - Lt) only depends on Lt and Xlf or XLt- and is an inhomogenous strong Markov process. Getoor and Sharpe [7] give related results. General and elegant treatments of last exit decompositions are given in Maisonneuve [18] and Pitman [21]. Kallenberg [14] proves that for Levy processes the fragments considered by Pittenger and Shih are conditionally independent given XL^ - and LF where the last exit time from a set F is an instance of a backward time. Under suitable conditions the laws of the two fragments are described. Another example of considering a decomposition of a Markov process at last exit time from 0 before and independent exponential time Sg is the proof of fluctuation equalities for Levy processes given by Greenwood and Pitman [9]. If X is a Levy process then it is known that the process reflected at the supremum defined by Yt — sups 0 which implies the Chapman-Kolmogorov equations for all s, t > 0 and x, y G E. M. Perman: A decomposition for Markov processes at an independent exponential time 53 To formulate the results the existence of a dual strong Markov process X on (E, E) relative to the measure £ will be assumed. This means that £ is an invariant measure for both X and X and Px(Xt e dy) = pt(x, y)£(dy) and Px(Xt e dy) = p(y, x)£(dy) (2.1) for all t > 0, x, y e E with pt(x, y) = pt(y, x). See [6] and the references therein for details. Denote by re(x, y) = / e_etp (x, y) dt and re (x,y)= / e_etpt(x,y) dt ./0 ./0 the resolvent densities of X and X respectively. For the sake of simplicity it will be assumed that X and X have infinite lifetimes Z under Px and Py for all x, y e E respectively. The assumptions on X imply that it is possible to define bridge laws PX,y (■)= Px(-|Xt = y) (2.2) for t > 0 and for x, y e E. By Proposition 1 in Fitzsimmons, Pitman, Yor [6] for any x, y e E and t > 0 with pt(x, y) > 0 there is a unique law PX,y on (Q, Ft) such that for any Fs-measurable functional F for 0 < s < t Et,y(F) ■ pt(x,y) = Ex (F ■ Pt_s(Xs,y)) , (2.3) where EX ,y and Ex are expectations with respect to measures PX,y and Px respectively. The laws Px,y provide a regular version of the family of conditional distributions P( |Xt = y). Furthermore by Corollary 1 in Fitzsimmons, Pitman, Yor [6] the law of the reversed bridge (X(t_s)_ : 0 < s < t) under Px,y has the law of the bridge of the dual process Py,x. The subject of this paper is the law of the process X started at a and run to an independent exponential time Se with rate 0 and conditioned on |XSe = b}. Conditionally on {Se = t, Xt = b} the law of the process will be the law of the bridge P^ b and the laws P,t b will serve as the regular version of the family of conditional distributions. Assume that a is a recurrent point of the process X. Let Ta = inf {t > 0: Xt_ = a or Xt = a}. Since a is assumed to be recurrent the assumptions imply that Pb(Ta < to) = 1 for all b e E. For t > 0 define the last exit time from a before time t as gt = sup{s < t: Xt_ = a or Xt = a} . Let (LJ1: t > 0) be the local time for the process X at a. We will assume that such a right continuous nondecreasing additive functional exists and only increases on the set M = {t > 0: Xt_ = a or Xt = a}. All the results will be valid for any choice of normalization of the local time. Let for s > 0 ts = inf{t > 0: > s} (2.4) be the right continuous inverse of the local time. From the strong Markov property of X it follows that (ts : s > 0) is a subordinator. Since we are assuming recurrence the local time at a will be unbounded and hence ts is well defined for all s. For simplicity we will assume that the set M has Lebesgue measure 0 almost surely. This means that the subordinator 54 Ars Math. Contemp. 12 (2017) 25-29 (ts : s > 0) has no drift. From properties of subordinators, see Bertoin [2], Ch. 3, it follows that Ea (e-0Tu) = e-^(e)u . (2.5) The notation X(t), La(t) and g(t) will be used for Xt, La and gt whenever necessary. The theorem to be proved is stated as follows. Theorem 2.1. Assume that a is a recurrent point for the process X and pt(a, b) > 0 for all t > 0. Let X0 = a and assume (La : t > 0) is the local time of X at a. If Sg is an exponential random variable with parameter 0 independent of X then, under the measure Pa: (i) The random variables La (Sg) and X(Sg) are independent with distributions Pa(La(Sg) € du) = ^(0) e-'^g)udu and Pa(X(Sg) € dy) = 0rg(a,y)dy (2.6) where ^(0) is the Laplace exponent defined in Equation 2.5. (ii) The processes (Xt: 0 < t < gse) and (XgSg +u :0 < u < Sg - gse) are independent. (iii) For bounded measurable Junctionals F and G Ea \F(Xs: 0 < s < tu) e-gTu 1 Ea [F(Xs :0 < s < gsg)| Lse = u] = a[ ( °E <_g<, u)-(2.7) Ea [e gTu ] Ea[G(X{se-s)- : 0 < s < Sg - gse)|Xs9 = b] = (2.8) Eb G(Xs: 0 < s < fa) e ...-gTa a) Eb[e-gTa ] where Eb refers to expectation under the law Pb of the dual process, and Ta = inf {t : Xt = a or Xt- = a} is the hitting time of a for X. The novelty lies in the fact that known special cases are covered by the more general Markov setup. Excursion arguments used are standard. 3 Excursion arguments Let n be a point process on an abstract space (S, S) with mean measure A. If n is a Poisson process then by Campbell's Theorem, see Kingman [16], p. 28, for any measurable f > 0 E (exp(—J f(x)n(dx)^ =exp J (1 - e-f (x)) A(dx)^ . (3.1) Conversely, if Equation 3.1 holds for any measurable f > 0, then n is a Poisson process with mean measure A. M. Perman: A decomposition for Markov processes at an independent exponential time 55 Assume that n is a Poisson process with mean measure A, and let h: S ^ [0, to) be a measurable function such that f (1 - e-h(x))A(dx) < to . J S (3.2) The random variable Sh defined by Sh = / h(x)n(dx) S is almost surely finite and non-negative with E (exp(-Sh)) = exp ^- (1 - e-h(x))A(dx)^ . (3.3) Define a new probability measure Q by dQ exp(-Sh ) dP E [exp(-Sh)] " (3.4) The following lemma is known in the literature, see Proposition 2.1 in James [12] and the discussion therein, or Proposition 2.4 in Bertoin [3]. Lemma 3.1. Under the measure Q, n is a Poisson process with mean measure e-h(x) • A(dx). Proof. It suffices to check that Equation 3.1 holds. Denote c = 1/E(exp(-Eh)) and let = fS f (x)n(dx) for a measurable funtion f > 0. One has Eq [exp(-Sf )] = cEP cEP exp (-Sh) • exp ^- J f (x)n(dx) ex^ -J (f (x) + h(x)) n(dx) ex^ -J (l - e-f (x)-h(x^ A(dx)^ = c exp (1 - e-f(x)) e-h(x) + (1 - e-h(x))l A(dx) ex^ -J (1 - e-f (x)) e-h(x) A(dx)^) =c S □ Let X be a cadlag strong Markov process. The set M = {t > 0: Xt- = a or Xt = a} is closed under the assumptions. Since we are assuming recurrence of X the complement of M is a countable union of bounded open intervals. The segments of the path of X on these open intervals are called the excursions of X away from the point a. By definition the open intervals coincide with the complement of the range of the subordinator (ts : s > 0) defined in Equation 2.4. Let Us be the space of cadlag paths 56 Ars Math. Contemp. 12 (2017) 25-29 w: [0, to) ^ E such that w(0) = a and there is a t > 0 such that for 0 < s < t we have w(s) = a, w(s-) = a, and w(t) = a or w(t-) = a and w(s) = S for s > t where S is the coffin state added to E. Let U be the a-algebra generated by the coordinate maps in Us. Define the point process (es: s > 0) of excursions of X in the sense of Ito as S if ts - rs- =0 (35) es(u) = XTs_+„ for u < ts - ts- and S else. ( . ' The process e is a Poisson process in the sense of Ito governed by the measure A x n where A is a multiple of the Lebesgue measure on [0, to) and n is the Ito excursion law. We can change the normalization of the local time, if needed, in order to ensure that A is the standard Lebesgue measure. See Rogers and Williams [25], Ch. 8 or Revuz and Yor [24], Ch. 8 for background on excursion theory. The connection between excursion theory and the law of a Markov process run up to an independent exponential time Sg is established through marking excursions. Let n be a Poisson process on an abstract space (S, S). If conditionally on n each point x e n is assigned a mark with probability p(x) independently of all the other points in n then the resulting marked and unmarked processes are both Poisson and are independent. If A is the mean measure of n the marked process will have mean measure p ■ A and the unmarked process (1 - p) ■ A. See Kingman [16], Ch. 5 for definitions and proofs. The excursion of the Markov process X straddling the independent exponential random time Sg can be interpreted as the first marked excursion of X where conditionally on e marks are assigned to an excursion e with probability 1 - e-gR(e) where R(e) stands for the duration of the excursion i.e. the length of the open interval of Mc containing Sg. Intuitively we can think that Sg is the first point in a Poisson process N on (0, to) with rate 0 and independent of X. Excursions straddling a point of N are considered marked and other excursions are considered unmarked. By independence properties of Poisson processes conditionally on e the marks are assigned independently and an excursion of length R(e) contains a point of the Poisson process with probability 1 -eR(e) which follows from the Poisson distribution of points contained in the excursion interval. See Sec. 49 in Rogers and Williams [25] for definitions and proofs. The following theorem is stated in slightly more general terms allowing the Poisson process (es: s > 0) to be killed at a rate q. This would correspond to excursions of X that have infinte length. The conclusions of the theorem are well known, see Sec. 49.4 in Rogers and Williams [25]. Theorem 3.2. Let (es : s > 0) be a possibly killed excursion process of X from a recurrent point a in the sense of Ito, and let Sg be an exponential random variable with parameter 0 independent of X. Denote by (La : t > 0) the local time process of X at a and (es: s > 0) the associated excursion process governed by A x n where A is the Lebesgue measure. (i) The local time La(Sg) during the excursion straddling Sg is an exponential random variable with parameter i (1 - e-gR(e)) n(de; R< to) + q ju where q is the rate of arrival of excursions with infinite lifetime. Moreover, it is independent of the excursion e* = e( LaSg) which may have infinite lifetime. M. Perman: A decomposition for Markov processes at an independent exponential time 57 (ii) Given La (gSg) = u the process of excursions (es : 0 < s < u) is a Poisson process in the sense oflto which is governed by the measure n given by n(de) = e-6R{e) n(de; R(e) < to) where n islto's excursion law and R(e) denotes the length of the excursion. Moreover, e* is conditionally independent of (es : 0 < s < u) given {La(So) = u}, and is independent of La(So). Proof. The first marked excursion in (es: s > 0) will arrive at an exponential time. The processes of finite length excursions and those of infinte length are independent so the rates of arrivals add. The rate of arrivals of marked finite length excursions is by definition equal to i (1 - e-0R(e)) n(de; R< to) . ju The two processes of marked and unmarked excursions are independent. This means that conditionally on La (So) = u the process (es : 0 < s < u) is a Poisson process on (0, u) x Us. The first marked excursion is picked according to the normalized law (1 - e-0R) • n irrespective of the local time La( So). This and the independence of marked and unmarked excursions conclude the proof. □ 4 Proofs Recall that under the assumptions on X and X and if pt(a, b) > 0 there is a measure Pa b corresponding to the bridge of X starting at a and conditioned to be b at time t. The family of Pt b is a family of regular conditional laws of X given Xt = b. If So is an exponential random variable of rate 0 then under Pa the family Pa b is a regular conditional law of X given {So = t,Xt = b}. The assumptions made on X and X also imply that X and X have no jumps at fixed times. See (3.18) in Getoor and Sharpe [8] for a proof. Let Tb = inf{t > 0: Xt = b or Xt- = b}. Assume further that Pb(Tb > 0) = 0 which in conjunction with right continuity and strong Markov property implies that XTb = b. Lemma 4.1. Assume that Pa(Tb < to) = 1. Assume that pt(a, b) > 0 for all t > 0. Then esn re(M) Pa(Tb e ds\X(Se) = b)= e- Pa(Tb e ds) re(a, b) ' where re (x, y) is the resolvent density. Proof. By assumptions on b and X we have Pa(Tb < So|X(So) = b) = 1. For fixed 0 < s 0 for y G B. Let 58 Ars Math. Contemp. 12 (2017) 111-126 F =\(Tb e ds). We have Pa(Tb e ds,Tb < Se,Xse e B) = = i 6e-et dt i pt(a,y)P*ay(Tb e ds) £(dy) js j B /CO r Oe-et dt j Pa (Tb e ds) pt-s (b, y) £(dy) = e-esPa(Tb e ds) f ady) i de-eupu(b,y) du J B J 0 = e-esPa(Tb e ds) dt re (b,y)£(dy) JB f re (b,y) Ib re (a,y) e-esPa(Tb e ds) i ■ dre (a,y) £(dy) jb re(a,y) It follows that esn rm re(b,b) Pa (Tb e ds\X (Se) = b)= e-esPa(Tb e du) re (a,b) ' □ Remark 4.2. Integrating the equation with respect to s over (0, to) in Lemma 4.1 gives the well known formula Ea(e^) = ^ . (4.2) rg(b, b) See e.g. Rogers and Williams [25], (50.7) on p. 293. Let us consider the process (Xt: 0 < t < Tb) given {XSe = b}. The following lemma gives the conditional distribution of this process given X(Sg) = b. Lemma 4.3. Assume that Pa(Tb < to) = 1. Assume thatpt(a, b) > 0 for all t > 0. Let F be a non-negative measurable functional of the process (Xt: 0 < t < Tb). Then Ea(F\X(Se) = b) = Ea (e-eTb ■ F) ■ ^^ , (4.3) re(a, b) where re (x, y) is the resolvent density. Proof. As in Lemma 4.1 we compute for an open neighbourhood B e E of b such that M. Perman: A decomposition for Markov processes at an independent exponential time 59 rg (a, y) > 0 and a bounded measurable functional F Ea [F •1(Tb < Sg, Tb e ds, Xse e B)] = = f 6e-gt dt f pt(a,y)E (F • 1(Tb e ds)) dy js j B = f 9e-gt dt f Ea (F • 1(Tb e ds)) pt-s(b,y) dy Js J B = e-gsEa (F • 1(Tb e ds)) f dy f 6e-gupu(b,y) du J B J 0 = e-gsEa (F • 1(Tb e ds)) 9 f rg(b,y) dy JB = e-gsEa (F • 1(Tb e ds)) f • 9rg(a,y) dy B rg(a, y) This in conjunction with the distribution of Tb from Lemma 4.1 completes the proof. □ The conclusions of Lemma 4.3 apply equally to the dual process X. Moreover under Pa the conditional law of the process (X(Sg-t)- : 0 < t < Sg) given XSg = b is equal to the law of X started at b run to an independent exponential time and conditioned to be a at the end. This implies that under the assumptions on a for any bounded functional G G^X(se-t)- : 0 < t < Sg - gs0^j XSg = b = Eb , (4.4) G (Xt: 0 < t < Ta) e-gf* 1 • V / J rg(b, a) because the last exit time gSg from a is the first hitting time of a for the reversed process. This is in accordance with Theorem 7.6 in Getoor and Sharpe [8] that excursions straddling a fixed time reversed and conditioned on the length are the excursions of the dual process. See also formula (3.12) in Ikeda, Nagasawa, Sato [11] who give the law of the process reversed from the lifetime of a killed Markov process. The case treated here considers killing at a constant rate. We are now in position to give the proof of Theorem 2.1. Proof. The second assertion in (i) is the definition of the resolvent density. Let (es: s > 0) be the excursion process of X from a. Marked excursions arrive at an exponential rate so we know that La(Sg) will be exponential. Since excursions are marked by an independent Poisson process, the event {La(Sg) > u} is equal to the event that there is no mark in the interval [0, tu], and has conditional probability e-gTu. Integration gives Pa(La(Sg) >u)= Ea (e-gTu) = e-u^(g), which by differentiation gives the density. To prove (ii) note that by Theorem 3.2 conditionally on {La(Sg) = u} the process of excursions (es: 0 < s < u) is independent of the excursion e* straddling Sg. Because marks to excursions are assigned by an independent Poisson process conditionally on R(e*) = r the mark is distributed at the distance U from the left endpoint with density 9e-gu/(1 - e-gr) Ea 60 Ars Math. Contemp. 12 (2017) 111-126 on [0, r] independently of the process of unmarked excursions and of the local time La (Sg). So the excursion e* together with the position of Sg within the duration of e* are independent of the process of unmarked excursions and of La(Sg). This proves the independence of the two processes in (ii). For the first assertion in (iii) note that conditionally on La(Sg) = u the excursions of (Xt: 0 < t < gSe) from the point a are a Poisson process with excursion law e-gR ■ n by Theorem 3.2 (i). On the other hand, if we let (es: 0 < s < u) be the Poisson process of excursions of X from a and choose h(e) = 6 R(e) in Lemma 3.1, under the new measure the process is still Poisson but with the mean measure e-Sh ■ n. But under the assumption that the set M has Lebesgue measure 0 we have = 6 ■ tu. The proof of the second formula in (iii) follows from Lemma 4.3 applied to the reversed process. □ Note the connection with Lemma 4.1 in Kallenberg [14] which states that for Levy processes with continuous densities Pa (F(Xs: 0 < s < Tu)|Tu = t) = Pa,a (F(Xs: 0 < s < t)|Lt = u) (4.5) where Pt refers to the law of the bridge of length t. Noting that Lemma 3.1 gives Pa (gs0 e dt|L(Sg) = u) = ^Effuf)dt) . (4.6) Ea (e gTu) Equations 4.5 and 4.6 imply part (ii) in Theorem 2.1. 5 Examples 5.1 Linear diffusions Let X be a regular diffusion on an interval I C R with speed measure m. It is well known that X has a jointly continuous density p(t, x, y) with respect to m: Px(Xt e A)= i p(t,x,y) m(dy) . (5.1) J A The density is symmetric in x and y which implies that for diffusions the dual process is the diffusion itself. Assume that the X has a recurrent point a and that the point a is not an atom of the speed measure m. This implies that the inverse local time at a has no drift. With such assumptions the conclusions of the Theorem 2.1 hold with X = X. Moreover, it is known that Eb (e-eT*) = and Ea (e-gT") = e-r» . (5.2) rg(a, a) See Rogers and Williams [25], Sec. 50. As the first example one can take X to be Brownian motion and a = 0. All the assumptions are satisfied. It follows that the two processes (Bt: 0 < t < gSe) and (BSe-t: 0 < t < Sg — gSg) are independent. For the first process we get s, , s E0 (F(Bt: 0 < t < TU)e-gTu) Eo (F(Bt: 0 < t < gse)|L(gs9) = l) = 0 1 ( ' ~~) U)-. (5.3) Eo (e gTu) M. Perman: A decomposition for Markov processes at an independent exponential time 61 It is well known that Eo (e-gT) = e-lV2g and Eb (e-gT) = exp (—|b|V28) . where T0 = inf{t > 0: Bt = 0}. The first assertion follows from (i) in Theorem 2.1 and the fact that Lt = |Bt|, see e.g. Revuz and Yor [24], p. 289. The hitting time distribution is an elementary consequence of the reflection principle for Brownian motion. The law of the second process, given BSg = a, is described by s, Eb (F(Bs: 0 < s < T0)e-gTo) E(G(Bse-t: 0 < t < Sg — gse )|Bse = b) = bl ( SE° (<-g<} 0)-1 . (5.4) which yields the result first obtained by Biane and Yor in [4]. See Leuridan [17] for an alternative elementary proof and Yen and Yor [28], Ch 9. for an alternative proof. Since the bridge laws for Brownian motion with drift B(m) = Bt + pt are exactly the same for all drifts the conditional law of (B(m) : 0 < t < Sg) given {B%? = b} does not depend on p. This means that Equations 5.3 and 5.4 hold for Brownian motion with drift with B instead of B(m) . The only change is that the resolvent density changes to that of Brownian motion with drift rg(a, b) 1 eM(b-a)-|b-a|^2g+M2 ^28 + p 2 The skew Brownian motion X(a) with parameter a e (0,1) is constructed by independently flipping the excursions of |Bt| up with probability a and down with probability 1 — a. A pair of dual processes with respect to Lebesgue measure are the processes X (a) and X(1-a). Both processes behave like Brownian motion away from 0 and the distribution of their local time at a fixed point is equal to the distribution of the local time of Brownian motion X(1/2). From the known transition densities of X(a), see p. 82 in Revuz and Yor [24], it follows rg(0, b) = (2a1(b > 0) + 2(1 — a)1(b < 0)) e-|b|72g . The skew Brownian motion satisfies all the assumptions made on the Markov process X. Equation 2.7 holds with the same E(e-gTu) as in the case of Brownian motion. Equation 2.8 holds with X replaced by standard Brownian motion started at 0. Let X be a Bessel process of dimension S e (0,2). Denote v = 5/2 — 1 e ( — 1,0). It is well known, see Revuz and Yor [24], that 0 is a recurrent point for X, satisfying all the assumptions and that the time X spends at 0 has Lebesgue measure 0. The results of Theorem 2.1 apply. Bessel processes are dual to themselves under the speed measure with density £(dx) = — ^^ dx on [0, to). Let Iv(z) and Kv(z) be the modified Bessel functions with index v. With respect to £ the transition density of X for a, b > 0 is given by 1 fab\ pt(a,b) = we 2t Habj. Using formula 15.55 in Oberhettinger [19] for 0 < a < b we get that 2 rg(b, a) = ^Iv(aV2d)Kv(bV2d) . av bv 62 Ars Math. Contemp. 12 (2017) 111-126 By Equation 4.2 ) = ( b y ' \ a) Kv (bV26) in accordance with Theorem 3.1 in Kent [15]. By Pa(T0 < to) = 1 and the continuity of paths we have Ta t T0 as a I 0. Letting a ^ 0 and taking into account that Kv (z) ~ (z/2)v for z ^ 0 we get Eb (e-ST) = (bV2^)V/2 Kv (bv^fl) . By Pitman, Barlow and Yor [1] there is a bicontinuous family of local times L of the process X such that Z't -i f (Xs)ds = - / bs-1Lbt db J 0 2 J 0 for bounded measurable functions f. With this choice the inverse local time is a stable subordinator of index —v with Laplace transform - ( -st ) ( 2!+ve-vr(i + v) \ Eo (e - )=—u —r—)—L). 5.2 Levy processes For a Levy process X the Lebesgue measure is invariant and the dual process is —X. If the process has continuous densities for t > 0, is recurrent and spends Lebesgue measure 0 at points the conclusions of Theorem 2.1 can be applied. An example is provided by symmetric stable processes of index a e (1, 2). These processes are recurrent and by scaling property the inverse local time is a subordinator of index 1 — 1/a. See Bertoin [2], Ch. 8. The independence of (Xt: 0 < t < gSe) and (Xgs +u: 0 < u < Ss —gSe) and scaling imply that given g1 the process (Xt: 0 < t < g1) is conditionally independent of (Xfl1+„: 0 < u < 1 — gi). This means that the two processes (Vgi : 0 < t < 0 and fXv+u(1-gi) :0 < u < ^ (5.5) Vvgi < < J V vr—gi < < J are independent. Scaling also implies that the inverse local time tu is a stable subordinator of index 1 — 1/a with E(e-dTu) = e-u^(s) = e-cSl 1/a for some constant c depending on the normalization of the local time. From Equation 2.7 we can compute E (e-(A+0K) E (e-A3S* lL(S) = u) = ¿(e-ST„) . (5.6) Using the form of ^(6) and unconditioning using Equation 2.6 gives ( 6 \1-1/a E (e-X'") = (AT?) . It follows that gSe ~ r(1 — 1/a, 6) and by independence Ss — gSe ~ r(1/a, 6). Using scaling again this gives the arc-sine law g1 ~ Beta(1 — 1/a, 1/a). This result is due to Chaumont [5]. See also Bertoin [2], p. 230. M. Perman: A decomposition for Markov processes at an independent exponential time 63 For another application let Y be a Levy process. Assume P(Y0 = 0) = 1 and define Yt = sups u} be the right continuous inverse of L. Denote by (eu : u > 0) the excursion process attached to the process X. It is in general not possible to reconstruct Y from the exursion process of the reflected process X. As noted by Greenwood and Pitman in their remark on p. 899 in [10], however, the process of excursions can be extended into a two dimensional Poisson point process such that X can be reconstructed. The idea is to add to the excursion at time u > 0 the jump of the ladder height process Hu = YTu. Denote Ju = Hu — Hu-. The resulting point process ((eu, Ju) : u > 0) is a Poisson point process in the sense of Ito in the space U x (0, to). Let Sg be an exponential random variable with rate 0 independent of Y. If the assumptions of Theorem 3.2 are met the following conclusions can be made: (i) The pairs of random variables (gSg, XSg ) and (Sg — gSg, YSg — YSg ) are independent. This follows from Theorem 2.1. (ii). The random pair (gg, Xg) is infinitely divisible. To prove this statement first recall a standard result about Levy processes: if Z is a d-dimensional Levy process and Sg is an independent exponential random variable, then the random variable (Sg, ZSg ) is infinitely divisible. See Bertoin, [2] p. 162. By Equation 2.8 applied to the reflected process X we find that the law of (gSg ,XSg ) given {Lg = u} is just like the sum of the points of the process ((Ru, Ju) : u > 0) where Ru is the excursion length at local time u and Ju is the jump of the ladder height process H. This last two-dimensional process is a map of the extended excursion process ((eu, Ju) : u > 0) and as such a Poisson point process on (0, to)2. Sums of Poisson processes are infinitely divisible so it follows that (gSg ,XSg ) given {LSg = u} is infinitely divisible. But LSg is exponentially distributed and infinite divisibility follows. The infinite divisibility of the pair (Sg — gSg, YPSg — YSg ) follows by duality arguments. See Lemma 9 in Bertoin [2], p. 164. The assertions about infinite divisibility and independence are true in general without additional assumptions on the reflected process X. See Greenwood and Pitman [9] for details. References [1] M. Barlow, J. Pitman and M. 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Williams, Path decomposition and continuity of local time for one-dimensional diffusions. I, Proc. London Math. Soc. (3) 28 (1974), 738-768. [28] J.-Y. Yen and M. Yor, Local times and excursion theory for Brownian motion, volume 2088 of Lecture Notes in Mathematics, Springer, Cham, 2013, doi:10.1007/978-3-319-01270-4, a tale of Wiener and Ito measures, http://dx.doi.org/10.1007/978-3-319-01270-4. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 67-76 Euler's divergent series and an elementary model in Statistical Physics Bill Allombert Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Liberation, F-33405 Talence Cedex, France Jean-Paul Allouche CNRS, Institut de Mathématiques de Jussieu-PRG, Université Pierre et Marie Curie, Case 247, 4 Place Jussieu, F-75252 Paris Cedex 05, France Michel Mendes France Institut de Mathematiques de Bordeaux UMR 5251, Université de Bordeaux, 351 cours de la Liberation, F-33405 Talence Cedex, France Received 3 April 2015, accepted 3 April 2016, published online 2 May 2016 We discuss the multiple integral of a multivariate exponential taken with respect either to the Lebesgue measure or to the discrete uniform Bernoulli measure. In the first case the integral is linked to Euler's everywhere divergent power series and its generalizations, while in the second case the integral is linked to a one-dimensional model of spin systems as encountered in physics. Keywords: Euler divergent series, Abel-Plana Formula, Stirling numbers, spin system, Ising chain. Math. Subj. Class.: 33E20, 28A35, 82B44, 82D30 1 Introduction Consider the integral (N > 1) E-mail addresses: bill.allombert@math.u-bordeaux.fr (Bill Allombert), jean-paul.allouche@imj-prg.fr (Jean-Paul Allouche), michel.mendes-france@math.u-bordeaux.fr (Michel Mendes France) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract (1.1) 68 Ars Math. Contemp. 12 (2017) 111-126 If ^ is the Lebesgue measure on (0, to)n and H = 1, the integral is linked to the series ]T( —1)"(n!)N-1x" n>0 which, for N = 2, is attributed to Euler. If N =1 the series reduces to (1 + x) 1 (convergent for |x| < 1) and for N > 2 it diverges for all x = 0. If on the other hand, ^ is the Bernoulli measure on the set {-1,1}N then the integral reads zn (x)=2n ^ exp i-h i- x n «£{±1}N \ V=1 / j = 1 and could represent a certain spin system described in Section 6. 2 Euler's divergent series If ^ is the Lebesgue measure on (0, to)n, we suppose that H > 0 and x > 0. There is no loss of generality in the choice H =1 in Formula 1.1; take new variables vj = Huj. Integrate ' N \ N \ N Zn(x) = / exp i — | Uj | — x J^J Uj | JJ^ duj J(0,)N \ \ I I with respect to duN to obtain Z ( ) i exp l—pN-1 Ujjj -1 d ZN(x) = W-i 1 + x Uj j0dUj * Suppose N > 2 since the case N = 1 is trivial. ZN converges for all complex x outside tN-1 W Uj the negative real axis (—to, 0). Expand + x nN=11 Uj) into a formal power series / n-1 \ N-1 N-1 Zn(x) = exp ( — £ Uj ) £ ( —1)"x" ^ Ujn n duj • A0,~)N 1 \ j=1 / n>0 j=1 j=1 If we accept to permute the summation with the integral, then N-1 „ Zn(x) = ^ ( —1)"x" n u" exp(—Uj)duj = ^ ( —1)"(n!)N-1x". n>0 j=1 A0,^ n>0 What happens if N =1 or 2? The case N =1 is trivial yet interesting, 1 Z1(x) = / exp(—u — xu)du = 0 1 + x Expanding the integral with respect to x we obtain Z1 (x) = ^n>0 ( —1)"x" and for x =1 we rediscover the well-known "equality" J2n>0 ( — 1)" = 1 • B. Allombert et al.: Euler's divergent series and an elementary model in Statistical Physics 69 exp(-u) du = 0.5963 ... and therefore concluded 0 1 + u ^ ( —1)n n! = 0.5963 ... n>0 a most astonishing equality! In his beautiful book G. H. Hardy [2] discusses in detail this case N = 2. Remark 2.1. The constant du = 0.5963... is called the Euler or the Euler- Gompertz constant (see [3], [1, Section 6.2], and in particular [1, Section 6.2.4] for the name "Gompertz"). Among the numerous results related to this constant we do not resist to write the following continued fraction expansion: exp(-u) 1 + u du 1 2 - 4 - 6 8 - '.. This continued fraction expansion is sometimes attributed to Stieltjes, but in [8] Stieltjes indicated that it was studied by Laguerre. We found indeed in [5, p. 154] that Laguerre considered e times the Prym function1 eQ(a) = ff" e1-xxa-1dx and obtained as consecutive approximations of eQ(0) the sequence 4 20 124 920 7940 7' 34 209' 2546 13327' which are exactly the values of the first few truncatures of the above continued fraction (also see Laguerre [4, p. 77]). Of course eQ(0) = fc ™ exp(-u) 1+u du = Z2(1): it would thus be interesting to obtain such nice continued fraction expansions for the quantities ZN(1). More generally, a formula given by Tannery in [9, p. 1699] or an easy rewriting of a formula given by Laguerre in [4, end of Page 75] reads -dt 1 x + 1 - x + 3 — x+5- But Z2(x) exp(—u) 1 r™ --u -du = — 1 + xu x + 7 — .. e u i r™ e-t e du = 1 e1/M —dt. x J o x + u h/a DO CXJ 1 0 4 9 x e t 1 4 9 oo t x 0 1Note that there seems to be a misprint in the formula given by Laguerre, where e1 x is replaced by e see the original definition by Prym [7, p. 169]. 70 Ars Math. Contemp. 12 (2017) 111-126 Hence Z2(x) 1 + x - 2 x2 4x 1 + 3x-- 9x2 1 + 5x-- 1 + 7x - '.. 3 The Borel operator The sequence can be defined recursively by means of the so-called Borel operator B : f ^ exp(-u)f(ux)du Jo -m)/(ux)du. o The Borel operator applies the series J2n>o f (n)(0) n onto J2n>0 f (n)(0)xn. Using the relation Z0 (x) = exp(-x) and ZN+1 = BZN, we see that the integral ZN is therefore the Nth iterate BN of x ^ exp(-x), or equivalently the (N - 1)st iterate BN-1 of x ^ (1 + x)-1. 4 The Abel-Plana summation and the r function In this section we study the behavior of ZN when N goes to infinity. Note that for real x > 0, the sequence N ^ ZN (x) is bounded from above by 1 and furthermore it is increasing. Indeed let AN(x) = ZN +1(x) - ZN (x) and nN(x) = x nN=1 M. Then An(x) = i exP I - ¿ u I ( * ( , - exP( nN (x))) n . J(0^)N \ J V1 + nN(x) Since ^ - exp(-t) > 0, ZN+1(x) > ZN(x) as claimed. Therefore ZN(x) tends to a limit which we now compute. Theorem 4.1. For all real x > 0, we have lim ZN(x) = 1. Proof. Since the result is trivial for x = 0, we may assume x > 0. We note that ZN (x) can be written as a diverging series Zn(x) = £ (-1)n/N(n) n>0 where fN : s ^ r(1 + s)N-1xs is an analytic function on the half-plane K(s) > -1. By blindly applying the Abel-Plana Formula (see [6, III, formula X]) to this series, we get f-1/2+iTO r(1 + z)N-1 xz ZN(x) = - -77—,-;- J-\/2-iw 2« sin(nz) _ fr(1/2 + it)N-1x-1/2+_^dt 2 cosh(nt) 1 2 B. Allombert et al.: Euler's divergent series and an elementary model in Statistical Physics 71 or by displacing the integration contour, I/2+íto p/1 + -1Xz Zn(x) = 1 - / + Z\ , dz (4.1) Ji/2-icx, 2, sm(nz) = 1 - f r(3/2 + ,t)N-V/2+" d( (4.2) ' 2 cosh(nt) v y ' —oo The convergence of the integrals is provided by the fact the r function decreases like exp(-72 |z|) as z goes to — l/2±iro (resp. 1/2±i), and sin(nz) increases like exp(n|z|). Strictly speaking, the Abel-Plana Theorem only applies for N = 0. However, by applying the Borel operator to the right-hand side and interverting the summations by Fubini's Theorem, we find that ír(i/2 + it)N-1x-1/2+¿í , Fn(x) = —' \ , ,-dt N ( ) J-n 2 cosh(nt) satisfies the same recursion as ZN(x). Indeed, . . r , ^ r(l/2 + it)N-1(lu)-1/2+it , , BFn (x) = exp(-u) ——----dtdu N ( ) Jo P( -J-to 2cosh(nt) rr(i/2 + it)N-1x-1/2+« r~ 1/2+it . w , = —-- u-1/2+it exp(-u)dudt. J-^ 2 cosh(nt) ./o From the identity r(1/2 + it) = J'0° u-1/2+lt exp(-u)du, it follows fr(1 /2 + it)Nx 1/2+it BFn (x) = i(1/ o+ w ^-dt = Fn+1(x) ./-rc 2 cosh(nt) therefore FN = ZN. Now since |r(3/2 + it) | < ^ < 1 for all t G R, the integral (4.2) converges to 0 when N goes to infinity for all real x > 0, thus we have proved: lim ZN(x) = 1. □ To conclude this section, we note that this formula for ZN involves a single integral which is much more suitable for numerical computations than the original formula involving a multiple integral. Note also that N need not be an integer. . . 5 A differential equation It might be worthwhile to mention that the function N \ N \ N ,JN (X) = / 72 Ars Math. Contemp. 12 (2017) 111-126 is a solution of a differential equation of order N - 1 with polynomial coefficients. Indeed, the shortest way to establish this is to introduce the linear operator U defined by U(z) = (xz)'. Clearly U(xn) = (n + 1)xn so that Uk(xn) = (n + 1)kxn. Then rN-1 ry /„,\ _ TTN- 1 V"^ / i\n/ i\N-In UN-1Zn(x) = UN(-1)n(n!)N-1x' n>0 = £ (-1)n(n!)N-1(n + 1)N- n>0 = £ ( —1)"((n +1)!)N-1xn ; n>0 xUN-1Zn(x) = £ ( —1)n((n +1)!)n-1xn+1 n>0 = 1 - Zn(x). The function ZN (x) is thus solution of the (N - 1)-st order differential equation xUN-1y + y = 1 with initial conditions y(0) = 1, y'(0) = -1,..., y(N-2)(0) = (-1)n-2 ((N - 2)!)N-1. The reader may well criticize the above proof since it involves divergent series. There is however no problem in justifying the result by applying the operator U to the integral representation of ZN (x); the calculations are just slightly more cumbersome. Example 5.1. Z2 (x), Z3 (x), Z4 (x) are respectively solution of the equations x2y' + (x + 1)y = 1 x3y" + 3x2y' + (x + 1)y = 1 y'" + 6x3y" + 7x2y' + (x + 1)y = 1 x y +oi y + 7x y + (x The reader will recognize the numbers above as the Stirling numbers of the second kind. This can be proved by noting that both families of numbers obey the formula «n+1,k = k&n,k + «n,k-1. 6 An unconventional spin system We now assume that ^ is the Bernoulli measure on {-1, +1}N: zn (x)=2n e exp i -h i ^ i- x n «£{±1}^ \ V=1 / j = 1 We interpret ZN as the partition function of a certain spin system which we describe below. Conventional spin systems are discussed for example in C. J. Thompson [10]. Imagine an N-component particle, each component of which has a spin uj = ±1, and which are instantaneously influenced by the N - 1 others. The "total" spin of the particle, B. Allombert et al.: Euler's divergent series and an elementary model in Statistical Physics 73 i.e., its sign is j uj. A real external field H acts on the spins. The Hamiltonian attached to the spin system in state u = (u^ u2,..., uN) with external field -H is then given by N N ^n uj+HJ2uj. j=1 i=1 The behavior of the spin system is controlled by the partition function, in particular by its thermodynamical limit log Zn (x) lim -—-- n^TO N Theorem 6.1. For all real x > 0, ZN(x) = cosh(x) cosh(H)N - (-1)N sinh(x) sinh(H)N. Proof. By using the relation exp(-1) = cosh(t) — sinh(t), we write zn(x)=2n ^ exp ( -h (^ uj illcosh(x n uj) - sinh(x n uj)). ue{±i}^ V \j=1 J J \ j=1 j=1 J Since f]f=1 uj = ±1, cosh is even and sinh is odd, it follows that 1 i 1 N \\ 1 N \ Zn (x) = 2N exp l -H l ^ uj l l l cosh(x) - sinh(x) JJ uj l . (6.1) m£{±1}^ \ \j=1 // V j=1 / The following two formulas are easily proved by recursion on N: £ exp IH uj )) = (2cosh(H))N m£{±1}I V \j=1 // £ exp lH uj im uj = (2sinh(H))N. ue{±1}N V \j=W / j=1 From Equation (6.1) it follows: Zn(x) = cosh(x) cosh(H)N - (-1)N sinh(x) sinh(H)N. □ Remark 6.2. Theorem 6.1 above implies that Zn(x) ~ cosh(H)N cosh(x), so that N ^-TO log Zn (x) lim -—-= logcosh(H) n^TO N z K ) which happens to be independent of x and which is continuous with respect to H. The system has no critical value of the external field and therefore presents no phase transition. 74 Ars Math. Contemp. 12 (2017) 111-126 7 A disturbed Ising chain In the preceding section we described an unconventional spin system. We now turn to the most familiar one, namely the one dimensional Ising chain (see [10]) with Hamiltonian N N j=i j=i where J is a "coupling constant". Actually this Hamiltonian corresponds to the parameters —H and - J but that makes no essential difference for our computation. We consider in fact a perturbed Ising chain with the additional term x\\\N=i uj. The Hamiltonian is therefore N N N H(u) = H ^^ Uj + Ujuj+i + x J^J j , uj+i + x|juj j=1 j=1 j=1 and the partition function is now Yn = 22N ^ exp(—H(u)) «e{±i}N which we propose to compute where we need to specify uN +1. Following most textbooks, we simplify the model by assuming that the chain is cyclic: un +i — ui. Theorem 7.1. Define A± — exp( —J) cosh(H) ± (exp(—2 J) cosh(H)2 + 2 sinh(2 J))2 , A± — exp( —J) sinh(H) ± (exp(—2 J) sinh(H)2 — 2 sinh(2 J)) 2 . Then Yn = 2N cosh(x)(AN + AN) — -tN^ sinh(x)(A\ + A-). Proof. Observe as in Section 6 that cosh x , sinh x „ yn = 2N yn 2N" yn where (N N \ — H ^ uj — J ^ ujuj+i ), j=i j=i / (N N \ N —Hj2uj—Jj2uj uj+i) n uj. j=i j=i j j=i The classical way to compute Y\ is to introduce the 2 x 2 transfer matrix L = i ¿i(l, 1) ¿i(l, —1) Li = 1 Li( —1,1) Li( —1, —1) B. Allombert et al.: Euler's divergent series and an elementary model in Statistical Physics 75 where Li(ui, «2) = exp ^ - §(« + «2) - Juiu^j . In other words j _ ( exp(-H — J) exp(J) Ll = ^ exp( J) exp(H — J) Then Yn = Li(MI,M2)LI(M2,U3) . ..LI(Un ,ui) «e{±i}N = ^ Lf (ui,ui) = Trace(Lf ) = Af + Aw «i£{±1} where A+ and A_ are the eigenvalues of L1, i.e., the solutions of A2 - 2Aexp(-J) cosh(H) + exp(-2J) - exp(2J) = 0. Therefore A± = exp(-J)cosh(H) ± (exp(-2J) cosh(H)2 + 2sinh(2J))2 The computation of Y^ is quite similar. Let L2(1, 1) L2(1, -1) where so that then ¿2 = l L2(-1,1) L2(-1,-1) H ¿2(^1,^2) = "i exp ( --^(ui + "2) - Jmi«2 L I exp(-H - J) exp( J) 2 - exp(J) - exp(H - J) Yx = L2(ui,M2)L2(u2,M3) . ..L2(u f,ui) «E{±1}W = ^ Lf (ui,ui) = Trace (L f )= Af + A- «i£{±i} where A+ and A_ are the eigenvalues of L2, i.e., the solutions of A2 + 2Aexp(-J) sinh(H) - exp(-2J) + exp(2J) = 0. Therefore A± = - exp(-J)sinh(H) ± (exp(-2J) sinh(H)2 - 2sinh(2J))2 . Finally Yf = cosh(x)(A+ + A f ) - sinh(x)(A+ + Af ). □ 76 Ars Math. Contemp. 12 (2017) 111-126 Remark 7.2. The reader will easily verify that for J = 0 we obtain the value of ZN computed in Section 6. Remark 7.3. It is not difficult to see that max{|A_|, |A_|, |A+|} < A+. Hence YN ~ 2N cosh(x)(A+) when N goes to infinity. This implies that the following limit exists, is continuous in both variables J and H, and is independent of x (as in Remark 6.2); therefore the system has no phase transition: y log YN A+ lim -zrz.- = log- N 2 =.oJ(exp(-J)cosh(H)+(exp(-2J )co8h(H)2+2(sinh(2J)) M. 8 Conclusion and acknowledgements This article illustrates a classical fact, namely that one formula may well lead to far distant and unexpected developments. Unifying themes is probably one of the most exciting aspects of mathematics. We warmly thank H. Cohen and A. Lasjaunias for their very kind help. We are very grateful to J.-Y. Yao and to the referees for their precise and constructive remarks and comments. J.-P. A. was partially supported by the ANR project ANR-12-IS01-0002 "FAN" (Fractals et Numeration). References [1] S. R. Finch, Mathematical constants, volume 94 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2003, doi:10.1017/CBO9780511550447, http://dx.doi.org/10.1017/CBO97 805115504 47. [2] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. [3] J. C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc. (N.S.) 50 (2013), 527-628, doi:10.1090/S0273-0979-2013-01423-X, http://dx.doi. org/10.1090/S027 3-097 9-2013-01423-X. [4] Laguerre, Sur l'integrale f° ^-x^, Bull. Soc. Math. France 7 (1879), 72-81, http://www. numdam.org/item?id=BSMF_18 7 9_7_72_1. [5] E. Laguerre, Sur la reduction en fractions continues d'une fraction qui satisfait a une equation differentielle lineaire du premier ordre dont les coefficients sont rationnels, J. Math. Pures Appl. 1 (1885), 135-166, http://eudml.org/doc/2 34 408. [6] E. Lindelof, Le calcul des résidus et ses applications a la théorie des fonctions, Gauthier-Villars, Imprimeur-Libraire, 1905, http://www.gutenberg.org/ebooks/2 97 81. [7] F. E. Prym, Zur Theorie der Gammafunction, J. Reine Angew. Math. 82 (1877), 165-172, doi: 10.1515/crll.1877.82.165, http://dx.doi.org/10.1515/crll.1877.82.165. [8] T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 8 (1894), J1-J122, http://www.numdam.org/item?id=AFST_18 9 4_1_8_4_ J1_0 . [9] J. Tannery, Sur les integrales euleriennes., C. R. Acad. Sci., Paris 94 (1882), 1698-1701. [10] C. J. Thompson, Mathematical statistical mechanics, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1972, a Series of Books in Applied Mathematics. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 77-87 The distinguishing index of the Cartesian product of finite graphs* Aleksandra Gorzkowska, Rafal Kalinowski, Monika Pilsniak Department of Discrete Mathematics, AGH University, Krakow, Poland Received 4 August 2015, accepted 2 March 2016, published online 20 May 2016 The distinguishing index D' (G) of a graph G is the least natural number d such that G has an edge colouring with d colours that is only preserved by the identity automorphism. In this paper we investigate the distinguishing index of the Cartesian product of connected finite graphs. We prove that for every k > 2, the k-th Cartesian power of a connected graph G has distinguishing index equal 2, with the only exception D'(Kf) = 3. We also prove that if G and H are connected graphs that satisfy the relation 2 < |G| < |H| < 2|g| (2"gII - 1) - |G| + 2, then D'(GdH) < 2 unless GUH = K22. Keywords: Edge colouring, symmetry breaking, distinguishing index, Cartesian product of graphs. Math. Subj. Class.: 05C15, 05E18 1 Introduction We use standard graph theory notation (cf. [6]). In particular, Aut(G) denotes the automorphism group of a graph G. An edge colouring breaks an automorphism p e Aut(G) if p does not preserve the colouring, i.e., there exists an edge of G that is mapped by p to an edge of different colour. The distinguishing index D'(G) of a graph G is the least natural number d such that G has an edge colouring with d colours that breaks all non-trivial automorphisms of G. Such a d-colouring is called distinguishing. This notion was introduced by Kalinowski and Pilsniak [10] as an analogue of the well-known distinguishing number D(G) of a graph G defined by Albertson and Collins [1] as the least number of colours in a vertex colouring that breaks all non-trivial automorphisms of G.1 As the distinguishing index is not defined for K2, we assume henceforth that K2 is not a connected component of any graph considered. *The research was partially supported by the Polish Ministry of Science and Higher Education. E-mail addresses: agorzkow@agh.edu.pl (Aleksandra Gorzkowska), kalinows@agh.edu.pl (Rafal Kalinowski), pilsniak@agh.edu.pl (Monika Pilsniak) 1 Fisher and Isaak [5] considered distinguishing edge colourings of complete bipartite graphs, but did not introduce any special notation or terminology. Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 78 Ars Math. Contemp. 12 (2017) 111-126 The distinguishing index of several examples of graphs was exhibited in [10]. For instance, D'(Pn) = D(Pn) = 2, for n > 3; D'(Cn) = D(Cn) = 2, for n > 6, and D'(Cn) = 3, for n = 3,4,5. There exist graphs G for which D'(G) < D(G), for instance D'(Kn) = D'(Kp,p) = 2, for any n > 6 and p > 4, while D(Kn) = n and D(Kpp) = p+1. It is also possible that D'(G) > D(G). All trees satisfying this inequality were characterized in [10]. The following general upper bound of the distinguishing index was proved in [10]. Theorem 1.1. [10] If G is a finite connected graph of order n > 3, then D'(G) < D(G) + 1. Moreover, if A is the maximum degree of G, then D' (G) < A unless G is a C3, C4 or C5. The distinguishing index was also investigated for infinite graphs [2] and their Cartesian product [3]. The Cartesian product of graphs G and H is a graph, denoted GUH, whose vertex set is V(G) x V(H). Two vertices (g, h) and (g', h') are adjacent if either g = g' and hh' G E(H), or gg' G E(G) and h = h'. Denote GdG by G2, and recursively define the k-th Cartesian power of G as Gk = GUGk-1. A non-trivial graph G is called prime if G = G1UG2 implies that either G1 or G2 is K1. It is easy to see that every non-trivial finite graph has a prime factorization with respect to the Cartesian product. For connected graphs it is also unique up to isomorphisms and the order of the factors, as has been shown by Sabidussi and Vizing (cf. [6]). Two graphs G and H are called relatively prime if K1 is the only common factor of G and H. Let v be a vertex of H. A Gv-layer (called also a horizontal layer of GUH) is the subgraph induced by the vertex set {(u, v) : u G V(G)}. An Hu-layer, or vertical layer, is defined analogously for a vertex u of G. Clearly, each horizontal layer is isomorphic to G and each vertical one is isomorphic to H. Therefore, speaking of a specified layer of GUH, we shall usually use only one coordinate of a vertex. The same refers to edges. We shall need knowledge of the structure of the automorphism group of the Cartesian product, which was determined by Imrich [7], and independently by Miller [11]. Theorem 1.2. [7, 11] Suppose ^ is an automorphism of a connected graph G with prime factor decomposition G = G1UG2U ... UGr. Then there is a permutation n of the set {1, 2,... ,r} and there are isomorphisms fa: Gn(i) ^ Gi, i = 1,... ,r, such that ^(X1,X2, ...,Xr ) = (fa (Xn(1)),fa(Xn(2)), . . .,fa (Xn(r))). It follows in particular that every automorphism of the Cartesian product of two relatively prime graphs is a composition of a permutation of vertical layers generated by an automorphism of G and a permutation of horizontal layers generated by an automorphism of H. For additional results about the Cartesian product consult [6]. Our main results are extensions of theorems about the distinguishing number of Cartesian powers and of Cartesian products of connected graphs to the distinguishing index. The results (and some of the proofs) are inspired by a paper [8] by Imrich, Jerebic and Klavzar. In Section 2 we generalize a result of Imrich and Klavzar. Theorem 1.3. [9] Let G be a connected graph and k > 2. Then D(Gk) =2 except for the graphs K2,K3, K whose distinguishing number is three. The second result that we extend is also due to Imrich and Klavzar: A. Gorzkowska et al.: The distinguishing index of the Cartesian product of finite graphs 79 Theorem 1.4. [9] Let G and H be connected, relatively prime graphs such that |G| < |H| < 2|g| — |G| + 1. Then D(GaH) < 2. In Section 3 we prove an analogous result (Theorem 3.4) for the distinguishing index of the Cartesian product of connected graphs, not necessarily relatively prime (let us note that, using our method of proof, Theorem 3.4 was already strengthened in [4] by omitting the assumption that G and H are relatively prime). We also obtain a slightly stronger result for trees (Theorem 3.1). In proofs, we usually use colours 1,..., d. If d = 2, then we also use colours 0 and 1, or alternatively red and blue. 2 Distinguishing Cartesian powers Let us start with the Cartesian powers of K2. Recall that the k-dimensional hypercube is isomorphic to Kk and denoted by Qk. As mentioned earlier, the distinguished index is not defined for K2 = Q1. Clearly, D'(Q2) = 3 since Q2 = C4. The following result was proved in [13]. Theorem 2.1. [13] If a graph G of order at least 7 contains a Hamiltonian path, then D'(G) < 2. Proposition 2.2. If k > 3, then D'(Qk) = 2. Proof. For k > 3 a hypercube Qk is Hamiltonian and has at least eight vertices. Therefore, D'(Qk) = 2 by Theorem 2.1. □ The distinguishing index of the square of cycles and for arbitrary powers of complete graphs with respect to the Cartesian, direct and strong products has been already considered by Pilsniak [12]. In particular, she proved that D'(Cm) =2 for m > 4, and D'(K^) = 2 for any n > 4 and k > 2. Here we consider Cartesian powers of arbitrary connected graphs. We first prove some lemmas. Lemma 2.3. Let G and H be connected, relatively prime graphs with D'(G) = D'(H) = 2. Then D'(GdH) = 2. Proof. We colour one G-layer and one H-layer with distinguishing 2-colourings. The remaining edges can be coloured arbitrarily. Such a colouring breaks all permutations of both horizontal and vertical layers. Since G and H are relatively prime, it follows from Theorem 1.2 that this colouring breaks all automorphisms of GdH. □ Lemma 2.4. Let G and H be two connected graphs where G is prime, |G| < ||H|| +1 and D'(H) = 2. Then D'(GdH) = 2. Proof. We first colour the H-layers of the graph GdH. There are at least two H-layers, so we colour all edges of one layer blue, all edges of another one with a distinguishing red-blue colouring. If there are more H-layers, then we colour them such that each of them has a different number of blue edges (including the H-layers coloured previously). This is possible since |G| < ||H|| + 1. Next, we colour all edges in every G-layer red. 80 Ars Math. Contemp. 12 (2017) 111-126 All automorphisms of the Cartesian product generated by the automorphisms of H are broken, since one H-layer assumes a distinguishing colouring. Also, no H-layers can be interchanged as every H-layer has different number of blue edges. If H has a factor H' isomorphic to G, then GdH has an automorphism interchanging H' with G. However, since all G-layers have only red edges and there exists an H-layer with only blue edges, such an automorphism does not preserve this colouring. □ Lemma 2.5. If H is a graph with 2 < D'(H) = d, then 2 < D'(HDK2) < d. Proof. We colour the edges of one H-layer with a distinguishing d-colouring, and all the edges of the other H-layer with the same colour, say 1. Next, we colour all edges of K2-layers with colour 2. Thus all automorphisms of the Cartesian product H□ K generated by the automorphisms of H are broken, since one of the H-layers assumes a distinguishing colouring. Also, the two H-layers cannot be interchanged as they have different numbers of edges coloured with 1. If H has a factor H' isomorphic to K2, then K2 □ H has an automorphism interchanging H' with K2. However, since all K2-layers have only colour 2 and there exists an H-layer with all edges coloured with 1, such an automorphism does not preserve the colouring. The equality for d = 2 is obvious since the prism of every graph has a non-trivial automorphism. □ We now consider the Cartesian powers of arbitrary connected graphs and continue with powers of connected prime graphs on at least three vertices. Lemma 2.6. If G is a connected prime graph with |G| > 3, then D'(Gfc) = 2 for every k > 2. Proof. The proof goes by induction on k. Let k = 2. There are n horizontal and n vertical layers, where n = |G|. Suppose first that G contains a cycle, i.e., ||G|| > n. We colour horizontal G-layers with two colours such that each of them has a different number of blue edges between 0 and n - 1. The other edges are coloured such that every vertical G-layer has a different number of blue edges between 1 to n. As every horizontal G-layer has a different number of blue edges they cannot be interchanged. The same is true for vertical G-layers. Therefore automorphisms of the Cartesian product generated by automorphisms of G are broken. Our colouring also breaks interchanging the factors, since there exists a completely red horizontal G-layer but no such vertical G-layer. Assume now that G is a tree. Every tree has either a central vertex or a central edge fixed by every automorphism. In case of a tree with a central vertex v, we colour the edges of G2 such that consecutive horizontal layers have 0,..., n -1 blue edges, and consecutive vertical layers have 0,..., n - 1 blue edges in such a way that the horizontal Gv-layer and the vertical Gv-layer have all edges coloured red and blue, respectively. The vertex (v, v) is fixed by every automorphism of G2, hence this colouring is distinguishing. If G has a central edge e0 = uv, we colour the edge (u, u)(v, u) red and the remaining three edges of the subgraph e0^e0 blue. The vertical and horizontal Gv -layers have all edges blue and red, respectively. The remaining edges of G2 are coloured as in the case of a tree with a A. Gorzkowska et al.: The distinguishing index of the Cartesian product of finite graphs 81 central vertex. Such colouring forbids exchange of the horizonal layers with the vertical layers. Thus D'(G2) = 2. For the induction step, we apply Lemma 2.4 by taking H = Gk-1 since |G| < ||Gfc"1H + 1. □ Let us now state the main theorem of this section that solves the problem of the distinguishing index of the k-th Cartesian power of a connected graph. Theorem 2.7. Let G be a connected graph and k > 2. Then D'(Gk) = 2 with the only exception: D'(K2) = 3. Proof. Let G = Gf □ Gp2 □ ... □ GPr, where p > 1, i = 1,..., r, be the prime factor decomposition of G. Assume first that Gj = K2, i = 1,2,..., r. Then for every i we have D'(Gkpi) = 2 due to Lemma 2.6. By repetitive application of Lemma 2.3 we get D'(Gk) = 2 since Gkpi and Gkpj are relatively prime if i = j. Suppose now that G has a factor isomorphic to K2, say Gf = K2. If pf > 2, then D'(K2kpi) = 2 and again D'(Gk) = 2 by Lemma 2.3 applied to K2kpi and Gp22 □ ... □ Gp-. The same argument applies in case p1 = 1 and k > 3. Finally, if p1 = 1 and k = 2 we apply Lemma 2.4 twice and we also get D'(Gk) = 2 unless r =1, i.e., Gk = K2. □ 3 Distinguishing Cartesian products In this section we investigate sufficient conditions on the sizes of factors of the Cartesian product of two graphs to have the distinguishing index equal to two. 3.1 Trees We begin with a result for trees. Observe first that, by Theorem 1.2, the Cartesian product of two non-isomorphic asymmetric trees is an asymmetric graph, so its distinguishing index is equal to 1. Theorem 3.1. Let Tm and Tn be trees of size m and n, respectively. If 2 < m < n < 22m+1 - + 1, then D'(TmDT„) < 2. Proof. If Tm is isomorphic to Tn, then the conclusion follows from Lemma2.6. Therefore, assume that Tm and Tn are non-isomorphic. Then they are relatively prime, and it is enough to prove the existence of a 2-colouring of edges of TmnT„ that breaks the automorphisms generated by automorphisms of Tm and those generated by automorphisms of Tn. In the proof we use the following well-known fact. In a rooted tree, if a parent vertex is fixed by every automorphism preserving a given colouring and its children cannot be permuted, then the children are also fixed. Assume first that n = 22m+1 - [m 1 + 1. We choose a root u0 of Tm as follows. If Tm has a central vertex, then we take it as a root u0. If Tm has a central edge, then we choose 82 Ars Math. Contemp. 12 (2017) 111-126 one of its end-vertices as u0 and the other one as ui. Then we choose an enumeration w0,..., um of vertices of the rooted tree Tm satisfying the following condition: if uj is the parent of Uj, then i < j. We enumerate the edge WjWj = ej. Thus E(Tm) = {e1,..., em}. Let v0 be a root of Tn chosen by the same rule as the root w0 of Tm. Then we analogously enumerate vertices and edges of Tn to obtain V(Tn) = {v0,..., vn}, E(Tn) = {ei,.. .,£„}. We begin by colouring the Tf -layer by putting colour 0 on the edges ej, for i = 1,..., [y], and colour 1 on the remaining edges of this layer. It is easy to see that we can choose such an enumeration of vertices, and hence of edges, that the root w0 is fixed by every automorphism of Tm preserving this colouring. Indeed, this is obvious if w0 is a central vertex; if ei = w0wi is a central edge of Tm, then we enumerate the vertices such that u1,..., u[mJ belong to the same subtree of Tm - e1, therefore our colouring breaks all automorphisms of Tm reversing the end-vertices of e1. Then, we similarly colour the TU -layer with 0 and 1 in such a way that the vertex (u0,v0) is fixed by every automorphism of TmDT„ preserving this partial colouring. Hence, the Tm -layer, as well as the -layer, is mapped onto itself by every ^ g Aut(TmDT„) preserving this colouring. Next, we colour the other layers. Consider the set S of all 22m+1 binary sequences of length 2m + 1. Each Tf -layer with i > 1 is assigned a distinct sequence sj = («0, «1,..., a,, 61..., b,) from S, where aj, j = 0,..., m, is the colour of the edge ej joining the vertex (wj, vj) with its parent in the Tu -layer (observe that a0 has been already defined for all i > 1), and bj, j = 1,..., m is the colour of the edge of the Tf -layer corresponding to ej. To break all permutations of Tn-layers we delete some sequences from the set S. First observe that the sum of each coordinate taken over all sequences in S is the same (and equal to 22m). Clearly, a T^3 -layer and a T„3' -layer cannot be permuted whenever j < [^ 1 < j' since the edges ej and ej' in the T,f -layer have different colours. Consider the set A = {sk g S : k = 1,..., [f 1 - 1}, where sk = (a0, a1,..., am, b1,..., bm) is a sequence such that «j = armHj = 1 j and all other elements of sk are equal to 0. Thus |S \ A| = 22m+1 - [f 1 + 1. We use the set S \ A to colour Tf -layers, i = 1,..., 22m+1 - [f 1 + 1, hence the numbers of edges coloured with 1 is distinct for every pair of Tn-layers that could be permuted. Thus, all edges in TmDT„ are coloured, and we obtain a distinguishing 2-colouring of TmDT„, when n = 22m+1 - [ f 1 + 1. Now, assume that the difference l = 22f+1 - [ f 1 +1 - n is positive. We have to choose l sequences from S \ A that will not be used in the colouring. To do this we apply the idea of complementary pairs used in [8]. Denote 0 = 1,1 = 0. A pair of sequences (ao, a^ . .., am, .. ., bm), (ao, a^ . .., am, .. ., b,) from S \ A is called complementary. When l is even, we choose | complementary pairs. When l is odd, we choose the sequence (0,..., 0) g S \ A and — complementary pairs. Again all permutations of layers in TmDT„ are broken by this colouring since for every A. Gorzkowska et al.: The distinguishing index of the Cartesian product of finite graphs 83 pair of Tn-layers that could be permuted, the numbers of edges coloured with 1 is distinct, because aj + aj =1, j = 1,..., m. □ The bound 22m+1 —" m 1+1 for the size of a larger tree is perhaps not sharp. However, it cannot be improved much since Proposition 3.2 below shows that the distinguishing index of the Cartesian product of a star K1n of size n and a path Pm of order m is greater than 2 whenever n > 22m+1. It also shows that the distinguishing index of the Cartesian product of two graphs with widely different orders and sizes can be arbitrarily large. Proposition 3.2. If m > 2 and n > 2, then D'(Ki,n npm} = \ 2m-yn\ unless m = 2 and n = r3 for some r. In the latter case D'(K1,nOP2} = r +1. Proof. Let d be a positive integer such that (d — 1}2m-1 < n < d2m-1. Denote by v the central vertex of the star K1jn, by v1,..., vn its pendant vertices, and by u1,..., um consecutive vertices of Pm. Suppose first that m > 3. Clearly, every automorphism of K1nnPm maps the Pm -layer onto itself. We colour the first edge of this layer with 1 and all other edges of it with 2. Thus the Pm-layer is fixed by every automorphism, hence the K1n-layers cannot be permuted. • • mmn Figure 1: A distinguishing 2-colouring of K1j32DP3 We want to show that the remaining edges of K1nnPm can be coloured in such a way that Pm-layers also cannot be interchanged. Then all non-trivial automorphisms of K1nnPm will be broken. Note that a colouring of the yet uncoloured edges can be fully described by defining a matrix M with 2m — 1 rows and n columns such that in the j-th column the initial m — 1 elements are colours of consecutive edges of the Pm! -layer, and the other m elements are colours of the edges of K1n-layers incident to consecutive vertices of the Pm! -layer. It is easily seen that there exists a permutation of Pm-layers preserving colours if and only if M contains at least two identical columns. There are exactly d2m-1 sequences of length 2m — 1 with elements from the set {1,..., d}, hence there exists a colouring with d colours such that every column of M is distinct. Therefore, D'(K1inDPm} < d = I" 1. On the other hand, n > (d — 1}2m-1 so for every edge (d — 1)-colouring of K1jnnPm, the corresponding matrix has to contain two identical columns, therefore D'(K1innPm) > d — 1. Figure 1 presents the case n = 32 and m = 3. For m = 2, we colour the edges of K1jnDP2 in the same way. The only difference is that every P2-layer has only one edge, hence the two K1n-layers need not be fixed. This 84 Ars Math. Contemp. 12 (2017) 111-126 is the case when n = d3, because then each element of {1,..., d}3 is a column in M, and there exists a permutation of columns of M which together with the transposition of rows of M defines a non-trivial automorphism of K1nnP2 preserving the colouring. Thus we need an additional colour for one edge in a K1n-layer. When n < d3, we put the sequence (1,1, 2) as the first column of M, and we do not use the sequence (1,2,1) any more, thus breaking the transposition of the K1jn-layers, and all automorphisms of K1jnnP2. □ Let us mention in passing that D'(K1inmCm) = [ , unless m < 5 and n = 22m. In the latter case D'(K1inmCm) = 2*^/n +1 = 3. The proof can be led on the lines of the proof of Proposition 3.2. 3.2 Arbitrary factors We now consider the Cartesian product of arbitrary connected graphs. We first formulate a result for relatively prime factors. Lemma 3.3. Let G and H be connected, relatively prime graphs such that 3 < |G| < |H| < 2|G| (2||g| - 1) - |G| + 2. Then D'(GdH) < 2. Proof. Let V(G) = {«1,..., u|G|}, E(G) = {eb ..., e||G||}, V(H) = {v1,..., v|H|}, E(H) = {e1,..., £||H||}. Assume that v1 is a root of a spanning tree TH of the graph H, and the vertices of H are enumerated according to the rooted tree TH, i.e., each child has an index greater than that of its parent. As G and H are relatively prime, the only automorphisms of GdH are permutations of G-layers and H-layers. We first colour the edges of the GV1 -layer with a sequence (&1,...,&||G||) = (1,..., 1). We shall not use this sequence to colour the edges of any other G-layer any more. Thus the GV1 -layer will be mapped onto itself by every automorphism of GdH preserving the colouring. From now on, we proceed similarly as in the proof of Theorem 3.1. For i = 2,..., n, the GVi -layer will be assigned a distinct sequence of colours («1, ... ,a|G|,&1,. .. ,b||G||), where « is a colour of the edge joining the vertex («, vj) to its parent in the rooted tree Th in the Huj-layer, and bj is a colour of ej in the GVi-layer. We have 2|G| (2||G| — 1) such sequences, as we excluded all sequences of the form (a!,...,« G|, 1,..., 1). Thus all permutations of G-layers are broken. To break permutations of H-layers, we also exclude all sequences sk = (a1,..., a|G|, b1,..., b|G|) with a1 = ... = ak = 1 and ak+1 = ... = a|G| = b1 = ... = b||G|| ^ 0, for every k = 1,..., |G| — 1. We have 2|G| (2|G| — 1) — (|G| — 1) sequences to colour |H| — 1 G-layers. Depending on the size of |H|, we also exclude a suitable number of complementary pairs of sequences («1,..., a|G|, b1,..., b||G||), (aT,...,«[Gj, b1,..., b|G|) A. Gorzkowska et al.: The distinguishing index of the Cartesian product of finite graphs 85 and, possibly, a sequence (0,..., 0). Thus we obtain a colouring of GDH with a set of sequences such that the number of 1's is distinct in any of the initial |G| coordinates. Therefore, no permutation of H-layers preserves this colouring. Hence, this is a distinguishing 2-colouring of GDH. □ Finally, we state the main result of this section. Theorem 3.4. Let G and H be connected graphs such that 2 < |G| < |H| < 2|G| (2"g" - 1) - |G| + 2. Then D'(GDH) < 2 unless G = H = K2. Proof. If G = K2, then |H| < 4. If H = K4, then either D'(H) =2 or H is a cycle or a star, and these cases were already settled in Section 2. To construct a distinguishing 2-colouring of K2DK4, colour one edge in one K4-layer and two adjacent edges in the other K4-layer red, and all remaining edges blue. Let |G| > 3. The case when G and H are relatively prime was settled by Lemma 3.3. Therefore, we focus here on the situation when G and H have at least one common factor. Then D'(GDH) > 2, since the automorphism group of GDH is non-trivial. Let G = Gk1 □ ... DG^4 and H = H^1 □ ... DH^ be the prime factor decompositions of G and H, respectively. Assume that the initial r factors are common, i.e., Gj = H for i = 1,..., r. Denote Gn = Gk1 □ ... DGkr, Hn = H1 □ ... DH^r. Thus G = G/ DG// and H = H/ DH//. We use the following notation n i = |G/1, n2 = |G//1, m 1 = |H/1, m2 = |H//1. We first show that the distinguishing index of the Cartesian product G// DH// = Gii + k1 D ... DG^r + kr is equal to 2. If G//DH// = K|, then |H/1 > 2 and the graphs G/DK2 and H/ satisfy the assumptions of Theorem 3.3, hence D'(GDH) = 2, unless |G/DK2| > |H/1, that is |H/1 < 4|G/1. In the latter case, we can also apply Theorem 3.3 for the graphs G/ and H/ which are relatively prime and satisfy the inequalities |G/1 < |H/1 < 2|Gi |(2^Gi H - 1) -|G/1 + 2 unless |G/1 = 2 and < |H/1 < 7, i.e., GDH = Kf DH/, where H/ is prime. So we can apply Proposition 2.2 and Lemma 2.4. In any case D'(GDH) = 2. If Gji+ki = K2 for every i = 1,..., r, then D'(Gj1+ki) = 2 due to Theorem 2.7, and hence D'(G//DH//) = 2 by repeated application of Lemma 2.3. If Gi1+k1 = K|, then analogously D'(G22+k2 D ... DG[r+kr) = 2, hence D'(G//DH//) = 2 by applying Lemma 2.5 twice. Consider now the graphs G' = G/ DG// DH// and H' = H/. Clearly, they are relatively prime and |H'| < |H| < 2|g| (2"g" - 1) - |G| + 2 < 2|g' 1 (2"g'" - 1) - |G'| + 2. 86 Ars Math. Contemp. 12 (2017) 111-126 If also |G'| = nin2m2 < mi = |H'|, then graphs G' and H' satisfy the conditions of Lemma 3.3, and consequently, D'(GDH) = D'(G'DH') = 2. Then suppose that nin2m2 > mi. We consider two cases here. Assume first that ni < n2m2, i.e., |G/1 < |G//UHn|. Hence, |G/1 < ||G//OHn|| + 1, and by repeated application of Lemma 2.4 we get D'(G') = 2. As |H '| < |G'|, we infer again from Lemma 2.4 that D'(GDH) = D'(G'DH') = 2. In the second case, i.e., when n2m2 < ni, suppose first that mi = |H/|< 2|Gi|(2i|Gi11 - 1) -|Gi| +2. Then D'(G/ DH/) < 2 since the assumptions of Lemma 3.3 are satisfied whenever |G/1 < |H/1. Recall that also D'(G//DH//) = 2 and graphs G/DH/ and G//DH// are relati\)ely prime. Hence D'(GDH) = 2 by Lemma 2.3. Otherwise, if mi > 2|G 1 (2I|Gi II - 1) -|G/1 + 2, then we arrive at the sequence of inequalities mi < nin2m2 < n2 < 2ni (2ni - 1) - ni + 2 < 2|Gi |(2|Gi 1 - l) - |G/1 +2 < mi, which is impossible. Then suppose that |G/1 = ni > mi = |H/1 (and still n2m2 < ni). Let G'' = G/ and H'' = G//DH/DH//. Clearly, |G''| < |H''| since |G| < |H|. Moreover, we have |H''| = n2m2mi < nimi < ni < 2^"' (2|G"" - 1) - |G''| + 2. It follows from Lemma 3.3 that D'(GDH) = D'(G''DH'') = 2. □ Acknowledgment. The authors are very indebted to Wilfried Imrich for introducing them to the concept of symmetry breaking in graphs, and for many helpful discussions and suggestions. References [1] M. O. Albertson and K. L. Collins, Symmetry breaking in graphs, Electron. J. 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PilSniak, Edge motion and the distinguishing index, preprint Nr MD 076, http://www. ii.uj.edu.pl/preMD. [13] M. PilSniak, Nordhaus-Gaddum bounds for the distinguishing index, preprint Nr MD 083, http://www.ii.uj.edu.pl/preMD. ¿^creative , ars mathematica ^commons contemporánea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 89-109 Xd(G), |Aut(G)|, and a variant of the Motion Lemma Niranjan Balachandran *, Sajith Padinhatteeri t Department of Mathematics, Indian Institute of Technology Bombay, India Received 13 May 2015, accepted 28 January 2016, published online 8 August 2016 Abstract The Distinguishing Chromatic Number of a graph G, denoted xD (G), was first defined in [5] as the minimum number of colors needed to properly color G such that no non-trivial automorphism ^ of the graph G fixes each color class of G. In this paper, 1. We prove a lemma that may be considered a variant of the Motion lemma of [15] and use this to give examples of several families of graphs which satisfy xD (G) = x(G) + 1. 2. We give an example of families of graphs that admit large automorphism groups in which every proper coloring is distinguishing. We also describe families of graphs with (relatively) very small automorphism groups which satisfy xD (G) = x(G) +1, for arbitrarily large values of x(G). 3. We describe non-trivial families of bipartite graphs that satisfy xD (G) > r for any positive integer r. Keywords: Distinguishing chromatic number, automorphism group of a graph, Motion Lemma, weak product of graphs. Math. Subj. Class.: 05C15, 05C25, 05C76, 05C80 1 Introduction For a graph G = (V, E) let us denote by Aut(G), its full automorphism group. A labeling of vertices of a graph G, h : V(G) ^ {1,..., r} is said to be distinguishing (or r-distinguishing) provided no nontrivial automorphism of the graph preserves all of the vertex * Supported by grant 12IRCCSG016, IRCC, IIT Bombay. tSupported by grant 09/087(0674)/2011-EMR-I, Council of Scientific & Industrial Research, India. E-mail addresses: niranj@math.iitb.ac.in (Niranjan Balachandran), sajith@math.iitb.ac.in (Sajith Padinhatteeri) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 90 Ars Math. Contemp. 12 (2017) 111-126 labels. The distinguishing number of the graph G, denoted by D(G), is the minimum r such that G has an r-distinguishing labeling (see [1]). Collins and Trenk introduced the notion of the Distinguishing Chromatic Number in [5], as the minimum number of colors r, needed to color the vertices of the graph so that the coloring is both proper and distinguishing. In other words, the Distinguishing Chromatic Number is the least integer r such that the vertex set can be partitioned into sets V1, V2,..., Vr such that each V is independent in G, and for every I = n e Aut(G) there exists some color class V such that n(Vj) = Vj. The problem of determining the distinguishing chromatic number of a graph G, or at least good bounds for it, has been one of considerable interest in recent times ([4, 13, 5, 2, 3]). Clearly, the notion of the distinguishing chromatic number begins to get more interesting only if the graph admits a large group of automorphisms, in which case, it can vary substantially from the usual chromatic number. It is known (see [5]) that xd (G) = |V | if and only if G is complete multipartite. Consequently, it is simple to see that there exist graphs G with x(G) = k,xD (G) = l+k,forany k,l, since for instance, a disjoint union of a clique of size k and Ki,; achieves the same. Some upper bounds for xd (G) (for instance, a version of Brooks' theorem for the distinguishing chromatic number) appear in [4], which also includes the inequality xd(G) < D(G)x(G). However, in many interesting large naturally occurring families of graphs, we have xd (G) < x(G) + 1 (see [3, 2, 5, 4]). In this paper, we seek to address three aspects of the problem of determining xd (G) for a given graph G. Firstly, we prove a lemma that may be considered a variant of what is now well known as the motion lemma, first introduced in [15]. The motion lemma basically says that if every nontrivial automorphism of a graph moves 'many' vertices then its distinguishing number is small. A similar lemma also appears in the context of graph endomorphisms and 'endomorphism breaking' in [12]. As a result of our variant of the Motion lemma, we give examples of several families of graphs G satisfying xd (G) = X(G) + 1. Secondly, we contrast the relation between the size of the automorphism group Aut(G) of a graph with its distinguishing chromatic number xd(G). A result describing an upper bound for xd(G) in terms of the prime factors of |Aut(G)| appears in [4]. However, our perspective is somewhat different. We demonstrate families of vertex transitive graphs G with large chromatic number, and xd (G) = x(G) + 1 even though |Aut(G)| is not very large (we have |Aut(G)| = O(|V|3/2)). As a contrast, we also demonstrate a family of graphs with arbitrarily large chromatic number, with 'super large' automorphism groups for which every proper coloring of G with x(G) colors is in fact distinguishing. This latter example also addresses a point raised in [3] and these contrasting results indicate that the relation between | Aut(G) | and xd (G) can tend to be haphazard. Finally, as we indicated earlier, while it is simple to give (the trivial) examples of graphs G with x(G) = r, xd (G) = r + s, for any r, s, non-trivial examples are a little harder to come by. Clearly, adding a copy (not necessarily disjoint) of a large complete multipartite graph to an arbitrary graph achieves this goal but we shall consider such examples 'trivial' since the reason for the blowing-up of the distinguishing chromatic number is trivially attributed to the presence of the complete multipartite component. While it seems simple to qualitatively ascribe the notion of what constitutes a nontrivial example in this context, we find it a bit tedious to describe it precisely. Our last result in this paper describes what we would like to believe constitutes a nontrivial family of bipartite graphs G such that xd(G) > r, for any r > 2. It turns out that large complete bipartite graphs do N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 91 appear as induced subgraphs in our examples, but that alone does not guarantee that the distinguishing chromatic number necessarily increases. Furthermore, what makes these nontrivial in our opinion, is the fact that the distinguishing chromatic number of these graphs is more than what one might initially guess. The rest of the paper is organized as follows. In Section 2, we state and prove what we regard as a variant of the motion lemma and use this to establish instances of families of graphs with xd (G) = x(G) + 1 in Section 3. In Section 4, we describe two families of graphs — G1 and G2 — with rather contrasting properties. For G e Gi, we have xd(G) = x(G) + 1 even though |Aut(G)| = O(|V|3/2); for G e G2, |Aut(G)| = w(e|V|) and yet every proper x(G) coloring of G is in fact distinguishing. In Section 5, we describe a family of bipartite graphs for which xd (G) > r, for any r > 2. The Section 6 contains some concluding remarks and open questions. 2 A Variant of the Motion Lemma Following [15], we recall that the motion of an automorphism ^ e Aut(G) is defined as m(4>) := {v e G : ^(v) = v} and the motion of a graph G is defined as m(G) := min m(^). 0GAut(G) 0=/ The Motion lemma of [15] states that for a graph G, if m(G) > 2log2 |Aut(G)| then G is 2-distinguishable. We prove a slightly more general criterion to obtain a similar conclusion for the distinguishing chromatic number. For a graph G with full automorphism group Aut(G), let G C Aut(G) be a subgroup of the automorphism group. For A e G and S C V(G) we define FixA(S) = {v e S : A(v) = v} and Fa(S) = |FixA(S)|.Let F(S) := max Fa(S). AgG A=/ Definition 2.1. The Orbit of a vertex v with respect to an automorphism A is the set OrbA(v) := {v, Av, A2v,... Ak-1v} where Ak v = v. Lemma 2.2 (A variant of the motion lemma). Let C be a proper coloring of the graph G with x(G) colors and let C1 be a color class in C. Let G be the subgroup of Aut(G) consisting of all automorphisms that fix the color class C1. For each A e G, let 0A denote the total number of distinct orbits induced by the automorphism A in the color class C1. If for some integer t > 2, f (G)= E teA-'Cl' 5. Proof. Firstly, we consider the case when q > 5 and q is prime and show that xd (LGq) < 3. Consider a 2-coloring of LGq by assigning color 1 to the point set P and color 2 to the line set L. It is easy to see that an automorphism of LGq that maps P into itself and L into itself corresponds to an automorphism of the underlying projective plane, and hence any such automorphism is necessarily in PGL(F^) (by the preceding remarks). In order to use Lemma 2.2, set G = PGL(F^) and observe that every A e PGL(F^), which is not the identity, fixes at most q + 2 points of LGq. Hence ñ . , 2 , (q2 + q + 1) - (q + 2) q2 + 2q + 3 eA < q + 2 +-2-=-2-. Consequently, f(c) < (q8 - q6 - q5 + q3) +, (31) f (G) <-t(g*+i)/2-+ 1 (3.1) Case 1: q > 7. For q = 7, t = 2, the right hand side of Equation (3.1) is approximately 1.16. Since the right hand side of inequality Equation (3.1) is monotonically decreasing in q, it follows that f (G) < 2 for q > 7, hence by Lemma 2.2, LGq admits a proper distinguishing 3-coloring. In particular, xd (LGq) = 3, for q > 7, since clearly, xd (LGq) > 2. Case 2: q = 5. In this case, for t = 2 we actually calculate f (G) using the open source Mathematics software SAGE to obtain f (G) ~ 1.2; see the Appendix for the code with relevant explanations. Again in this case, xd (LG5) = 3. We also can prove that xd(LG2) = 4 and xd(LG3) < 5; these proofs are included in the Appendix for the sake of completeness. If q = pn for n > 2 and a prime p, we note that the cardinality of the automorphism group of PG(2, Fq) equals n|PGL(F3)| < log2(q)|PGL(F3)|. As in the prime case, we have f (g) < log2 q(q8 - q6 - q5 + q3) +1 t For q = 8 and t = 2 the right hand side is approximately 1.01. By the same arguments as in the preceding section, it follows that xd (LGq) = 3. □ 1 It follows that this group is contained in the full automorphism group. The full group is larger since it also includes maps induced by isomorphism of the projective plane with its dual. 94 Ars Math. Contemp. 12 (2017) 111-126 For q = 4 we calculate f (G) ~ 1.2. for q = 4, and t = 3 using the same SAGE code in the case q = 5 to make the actual computation, so we have xd (LG4) < 4. We believe that xd (LG4) = 3 though our methods fall short of proving this. 3.2 Levi graphs of order one Suppose n, k G N and 2k < n, consider the bipartite graphs G = G(L, R, E) where L := (¿1) corresponds to the set of k - 1 subsets of [n], R := ([k]) corresponds to the k subsets of [n], and u G L, v G R are adjacent if and only if u c v. We shall refer to these graphs as Levi Graphs of order one and we shall denote them by LGi(k, n), or sometimes, simply LG1. Note that for each u G L, v G R we have d(u) = n — k + 1 and d(v) = k. It is clear that Sn c Aut(LG1). But in fact Aut(LG1) = Sn, and this is a fairly routine exercise, so we skip these details. We shall use Lemma 2.2 to determine the distinguishing chromatic number of LG1(k, n). Following the notation of the lemma, set Fa := {v G R : a(v) = v} for a G Sn and let F = max FI. a=I Lemma 3.2. For n > 4, F < (^-2) + 2) and equality is attained if and only if a is a transposition (ij) for some i = j. Proof: Firstly, it is easy to see that if a = (12) then |FCT | = (k—2) + (n—2), so it suffices to show that for any n that is not of the above form, | Fn | < | Fa |. Suppose not, i.e., suppose n G Sn is not an involution and |Fn | is maximum. Write n = O1O2... Ot as a product of disjoint cycles with |O1| > |O2| > • • • > |Ot|. Then either |O11 > 2, or |O11 = |O21 = 2. If |O11 > 2, then suppose without loss of generality, let O1 = (123 • • •) If h G Fn then either {1,2} c h or {1,2} n h = 0. In either case we observe that h G Fa as well. Therefore Fn C Fa. Furthermore, note that a fixes the set g = {1, 2,4,..., k + 1}, while n does not. Hence |FCT | > |Fn|, contradicting that |Fn| is maximum. If |O11 = |O21 = 2, again without loss of generality let O1 = (12), O2 = (34). Again, h G Fn implies that either {1, 2} c h or {1, 2} n h = 0, so once again, h G Fa ^ h G Fn. Furthermore, {1, 2, 3,5,..., k + 1} G Fa n Fn, which contradicts the maximality of |Fn |. □ For k > 2 define n0(k) := 2k +1 for k > 3 and n0(2) := 6. Theorem 3.3. xd(LG1(k, n)) = 3 for k > 2 for n > n0(k). Proof. We deal with the cases k = 2, k = 3 first, and then consider the general case of k > 3. For k = 2, let A = {(1, 2), (2,3), (2,4), (3,4), (4,5), (5,6),..., (n — 1, n)}, and consider the coloring with the color classes being L, A, R \ A. Consider the graph G with V(G) = [n] and E(G) = A. Observe that the only automorphism G admits is the identity. Since a nontrivial automorphism that preserves all the color classes of this coloring must in fact be a nontrivial automorphism of G, it follows that the coloring described is indeed distinguishing. If k = 3, note that the coloring described by the sets R, A, L \ A is proper and distinguishing for the very same reason. N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 95 For the case k > 4, we use Lemma 2.2 with t = 2 and G = Aut(LG1). From Lemma 3.2 we have F < ("I2) + ("k2). Let C = R be the color class to be parted randomly and assign color 3 to all vertices in L = (fc["1). Then we have, f(G) < |Aut(LGi)|21 (F-|Cl|) + 1, (3.2) where |C| = (£). Therefore from Equation 3.2, we have n! f (G) < 2Kc+1, ( n)_( n — 2)_( n — 2) where K = (k) (k—2 ( k ). For n > 2k it is not hard to show that 2"jt < 1 for n > n0(k), so, by Lemma 2.2 we are through. □ 3.3 Weak product of graphs The distinguishing chromatic number of a Cartesian product of graphs has been studied in [3]. The fact that any graph can be uniquely (upto a permutation of the factors) factorized into prime graphs with respect to the Cartesian product plays a pivotal role in determining the full automorphism group. In contrast, an analogous theorem for the weak product only holds under certain restrictions. In this subsection, we consider the n-fold weak product of certain graphs and consider the problem of determining their distinguishing chromatic number. To recall the definition again, the weak product (or Direct product as it is sometimes called) of graphs G, H denoted G x H, is defined as follows: V(G x H) = V(G) x V(H). Vertices (g1, h1), (g2, h2) are adjacent if and only if {g1,g2} € E(G) and {h1, h2} G E(H). We first collect a few basic results on the weak product of graphs following [9]. For more details we refer the interested reader to the aforementioned handbook. Define an equivalence relation R on V(G) by setting xRy if and only if N(x) = N(y) where N(x) denotes the set of neighbors of x. A graph G is said to be R - thin if each equivalence class of R is a singleton, i.e., no distinct x, y G V(G) have the same set of neighbors. A graph G is prime with respect to the weak product, or simply prime, if it is nontrivial and G = G1 x G2 implies that either G1 or G2 equals Kf, where Kf is a single vertex with a loop on it. Observe that Kf x G = G. Before we state our main theorem of this subsection, we state two useful results regarding the weak product of graphs. If G is connected, nontrivial, and non-bipartite then the same holds for Gxn. This is a simple consequence of a theorem of Weischel (see [9] for more details). The other useful result is the following theorem which also appears in [9]. Theorem 3.4. Suppose $ is an automorphism of a connected nonbipartite R — thin graph G that has a prime factorization G = G1 x G2 x ... x Gk . Then there exist a permutation n of {1,2,..., k}, together with isomorphisms fa : Gn(j) ^ Gj, such that ^¡r^ xk ) = ($1(xn(1) ^ . . . , (xn(k))). We are now in a position to state our main result regarding the distinguishing chromatic number for a weak product of prime graphs. An analogous result for the cartesian product of graphs, under milder assumptions, appears in [3]. 96 Ars Math. Contemp. 12 (2017) 111-126 Theorem 3.5. Let G be a connected, nonbipartite, R — thin, prime graph on at least 3 n-times /-*-s vertices. Denote by Gxn the n-foldproduct of G, i.e., Gxn := G x G x ... x G. Suppose further that G admits a proper x(G) coloring with a color class C1 such that no non-trivial automorphism of G fixes every vertex of C1. Then xd(Gxn) < x(G) + 1 for n > 4. Proof. Let G be connected, non-bipartite, R — thin, and prime. We first claim that Aut(Gxn) = Aut(G) I Sn, the wreath product of Aut(G) and Sn. To see this, note that if G is R — thin one can easily check that Gxn is also R — thin. Moreover since every connected non-bipartite nontrivial graph admits a unique prime factorization for the weak product (see [9]), it is a simple application of Theorem 3.4 to see that Aut(Gxn) = Aut(G) I Sn. This proves the claim. Suppose x(G) = r and let [Ci : i e [r]} be a proper coloring of G. Then Ci x Gxn-1,i e [r] is a proper r coloring of the graph Gxn, so x(Gxn) < r. On the other hand, the map g ^ (g,g... ,g) is a graph embedding of G in Gxn, so x(Gxn) = r. Let us denote the aforementioned color classes of Gxn by C',i e [r]. We claim that xD (Gxn) < r + 1 and show this as a consequence of Lemma 2.2. By hypothesis there exist a color class, say C1 in G such that no nontrivial automorphism fixes each v e C1. Consider C' = Ci x Gxn- 1 and for each element in C' assign a value from [1,r + 1} uniformly and independently at random. This describes a proper (r + 1) —coloring of Gxn. By Lemma 2.2, we have F — T f (G) < n!IAut(G)ln2 — + 1 (3.3) where T = |C1 x Gxn- 11, G = n!IAut(G)ln and F is as in Lemma 2.2. Claim: If there exists a nontrivial automorphism of Gxn which fixes each color class C',i = 1... ,r, then it cannot also fix each vertex of C'. To prove the claim, suppose ^ is an automorphism of Gxn which fixes C' for each i e [r], and also fixes C' point-wise. By Theorem 3.4, there exist $ 1, $2,... ,$n e Aut(G) and n e Sn such that ^{x i,X2,...,Xn) = ($ 1 {xn( 1)),$2(xn(2)),...,$n(xn(n})) (3.4) for all (x 1,x2,..., xn) e Gxn. Now note that if ^ fixes C' point-wise then $ 1 fixes C1 point-wise. Indeed, 4>(x1, x2, . . . , xn) = (x1,x2 ,...,xn) ^^ {$1(xn(1)),$2(xn(2}),. . .,$n(xn(n})) = (x1,x2 , . . . , xri) ^^ $i(xn(i}) = xi forall i e [r]. (3.5) Since Equation (3.5) holds for all vertices (x1, x2,..., xn) e Gxn with x1 e C1 and xi e G , 2 < i < n, we conclude that n = I, $i = I, for 2 < i < n, and $1 acts trivially on C1. But then by the hypothesis on G, it follows that $1 = I in G and hence ^ = I. We now show that F < (|C11 — 2)IGIn-1. We adopt similar notations as in Lemma 2.2 and for simplicity, let us denote |G| = m. For ^ e Aut(Gxn) we shall write ^ = ($1,$2,... ,$n : n) to denote the map , x2 , . . . , xn ) = ($1(x^(1)), $2(x^(2)), ..., $n(xn(n})) N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 97 as in Equation (3.4) (see Theorem 3.4). Suppose ^ fixes the vertex (x^x2,..., xn) G Gxn. In particular we have x^) = (xj) for all i. It then follows that for all k, we have (Xnfc(i)) = xnk+1 (i) for each i. Consequently, if n has t cycles in its disjoint cycle representation then ^ can fix at most |C1|mt-1 vertices in C1. If n = I, then t < n, and in this case, since m > 3, n > 4, we have |C1|mt-1 < (|C11 — 2)mn-1. If n = I, then ^ is non-trivial if and only if = I for some i. In this case ^i(xi) = xi for all i, so (x1, x2,..., xn) is fixed by ^ if and only if xi G Fix^ for n all i. Consequently, = f] F^. Observe that if ^ is not a transposition then it moves i=1 at least three vertices, say x, y and z in G. In particular, ^ does not fix any vertex of the form (x1, x2,..., g,..., xn), where g G {x, y, z} and appears in the ith position. Thus, it follows that F^ < |C1|mn-2(m — 3). If is a transposition for some i > 1 then it is easy to see that F^, < (|C11 — 3)mn-1 < (|C1| — 2)mn-1. Finally, if is a transposition, then again F < (|C1| — 2)mn-1. This proves the claim. Setting F = (|C11 — 2)mn-1 , T = |C1 |mn-1 in Equation (3.3) gives us n!|Aut(G)|n +1 J(G) < -2m-1--+ 1 It is a simple calculation to see that the first term in the above expression is less than 1 for all m > 3 and n > 4. This completes the proof. □ Corollary 3.6. xd (Krxn) = r + 1 for n > 4, and r > 3. Proof. First note that for r > 3, Kr is prime, non-bipartite, and R — thin. Hence by Theorem 3.5 it follows that xd(Krxn) < r +1. A result of Greenwell and Lovisz (see [8]) tells us that all proper r—colorings for Krxn are induced by colorings of the factors Kr. In particular, it implies that xd (Krxn) > r. □ 4 xd(G) versus \Aut(G)\ As indicated in the introduction, one aspect of the problem of the distinguishing chromatic number of particular interest is the contrasting behavior of the distinguishing chromatic number vis-i-vis the size of the automorphism group. Our sense of contrast here is to describe the size of the automorphism group as a function of the order of the graph. First, note that one can give somewhat trivial examples of graphs with xd (G) > x(G) > k for any k and with a very small automorphism group as follows. Start with an arbitrary rigid graph, i.e. a graph with no non-trivial automorphism, with chromatic number larger than k. Now fix an edge e = {x, y} and 'blow up' the vertices x, y by small disjoint subsets X, Y respectively, and replace the edge e by the complete bipartite graph on the sets X, Y. The new graph satisfies xd (G) > x(G) rather trivially and one can ensure that by picking small subsets X, Y we can ensure that the full automorphism group is not too large either. 98 Ars Math. Contemp. 12 (2017) 111-126 In some sense, these examples are not very interesting because the fact that the distinguishing chromatic number exceeds the chromatic number for these graphs is attributable to a 'local' reason. It however becomes a more intriguing problem if we insist that the graph is also vertex transitive. Our first theorem in this section gives examples of vertex transitive graphs that admit 'small' automorphism groups, and yet have xd (G) > x(G) and with arbitrarily large values of x(G). Theorem 4.1. Given k G N, there exists a sequence of graphs Gni satisfying 1. X(Gni) > k, 2. XD (Gni ) > X(Gni ), 3. Gni is vertex transitive and |Aut(Gni )| = O(n3/2). Proof. Let q > k be prime and suppose S c Fq is a subset of size . We define the graph GS as follows: The vertices of V(GS) are the points of the affine plane AG(2, q); u = (ui, u2) and v = (vi, v2) are adjacent in GS if and only if v1 = u and (v2 - u2)(vi -mi)-1 G S. We denote (v2 - u2)(vi - ui)-i by s(u, v). For a, P G Fq, consider the set if := {(P + x, P + xa) : x G Fq}. We shall call the sets if as lines in what follows. Observe that, for each a G S and P G Fq, the sets is a clique of size q, so x(Gs) > q. We shall denote the independent sets2 {(P, x + P) : x g Fq} by Similarly, if a G S the set is an independent set of size q, the collection {/f : P g Fq} describes a proper q-coloring of GS, hence x(Gs) = q. Claim: xd(GS) > q. Let G = {G1, C2,..., Gq} be aproper q-coloring of GS. We claim that each Gj is a line, i.e., for each 1 < i < q we have Gj = for some a G S, P G Fq. Observe that for a G S, the collection C = {/f |P G Fq} partitions the vertex set of GS into cliques of size q. Therefore, in any proper q-coloring of GS, each color class contains exactly q vertices. Next, we recall a result of R6dei [14] which states that for a prime number q if X c AG(2, q) such that |X| = q and X is not a line then the set S(X) = {s(x, y)|x = y, x, y G X } has size at least . If a color class Gj is not a line then by the theorem of R6dei, | S(Cj) | > (q++3) and since |S| = 2—1 this implies that S(Gj) n S = 0. But then this contradicts that Cj is independent in Gs . In particular, any proper q-coloring G of GS must be a partition of the form {/f : P G Fq} with a G (Fq U {to}) \ S. Then the map ^a(x,y) = (x + 1, y + a) if a = to, ^~(x,y) = (x, y + 1) is a nontrivial automorphism that fixes each color class of G. This establishes that XD (GS) > q and proves the claim. Now, we shall show that for a suitable choice of S, GS has a relatively small automorphism group. Our choice of subset S shall be a uniformly random subset of Fq. Note that our earlier proof of the claim in fact shows that any maximum independent set corresponds to a line in AG(2, q). We now make the observation that all maximum sized 2 These are independent in Gs since ^ (/ Fq N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 99 cliques also correspond to certain lines in AG(2, q). Indeed, suppose X is a maximum clique of size q which does not correspond to a line. Again, by R6dei's theorem we have |S(X)| > . Since X is a clique, S(X) c S, but this contradicts the fact that |S| = . Consequently, if $ G Aut(GS) then since maximum cliques (respectively, maximum independent sets) are mapped into maximum cliques (resp. maximum independent sets), it follows that $ is a bijective map on F^ which maps affine lines into affine lines in AG(2, q) (as a consequence of [14]). Hence, it follows that Aut(GS) c AGL(2, q) (see [11]). In other words, any $ G Aut(GS) can be written as A + b for some A G Aut0(GS) and b(= $(0,0)) G F^, where Aut0(GS) c Aut(GS) is the subgroup of automorphisms which fix the vertex (0,0) G V(GS). The following lemma shows that for a random choice of the set S, the automorphism group Aut(GS) is not very large. Lemma 4.2. Suppose S is picked uniformly at random from the set of all subsets of Fq. Then asymptotically almost surely, Aut(GS) = {XI + b : X G F*, b G V(GS)}. Consequently, IAut(GS)| = q2(q — 1) asymptotically almost surely. Here by the phrase asymptotically almost surely we mean that the probability that Aut(GS) = {XI + b : X G F**,b G V(GS)} approaches 1 as q ^ to. "ul ^ »«im y) — ^i,-"! y) i v^ij ^2/ ^i, g ^ q m.v. ^A g Aut0(GS). Here, A G GL(2, q) corresponds to a matrix ^ ^ for a, b,c,d G Fq with Proof. Since we have already observed that Aut(GS) c AGL(2, q), every $ G Aut(GS) can be written in the form $(x, y) = A(x, y) + (bi, b2) for some bi,b2 G Fq and A G ab ad — bc = 0. We introduce the symbol to and adopt the convention that a + to = to, a to = to for a = 0, and ^ = to for a = 0. For $ G Aut(GS), define a map f : Fq U{to} ^ Fq U{to} as follows: da + c a f(a) = —~T, if a = — I, a + ba b ^ = to, Mto) = Observe that f is trivial if and only if b = c = 0 and a = d. Let x = (xi, x2),y = (yi,y2) be two adjacent vertices in GS. Since $(x) is adjacent to $(y), we have c(yi — xi) + d(y2 — x2) d ■ s(x,y) + c s($(x),$(y)) = —,-r—r7-r = T--r-—. a(yi — xi) + b(y2 — x2) b ■ s(x,y) + a Observe that yi — xi is nonzero since s(x, y) G S. Therefore we have, s($(x),$(y))= U(s(x,y)). (4.1) Also note that if $ = XI then for m G Fq and k G N, setting fik)(M) := U ◦ f ◦■■■◦ U(M = M v-v-' k-fold 100 Ars Math. Contemp. 12 (2017) 111-126 yields a quadratic equation in m, so there are at most two values of m G Fq satisfying /^(m) = M. In other words, for each positive integer k, the map / admits at most two orbits of size k. Moreover if A G Awt(GS) then by equation (4.1), /A(S) = S. Consider the event E: There exist a nontrivial automorphism A G Awt0 (GS) such that /A is not the identity map. Observe that E is the union of the events EA where the event EA is described as follows: For any A g GL(2, q) where A = A/, A = 0, S is the union of /A orbits. Recall that /A is not the trivial map if and only if A = A/ for any A = 0. By a favorable automorphism, we shall mean an automorphism A G Awt0(GS), A = A/ such that S is union of /A orbits. By the preceding discussion, it follows that a favorable automorphism of GS induces a partition A of ^r1 in which there are at most two parts of any size. Therefore the number of favorable automorphisms is at most twice the number of integer partition of ^^ in which there are at most two parts of any size which is clearly less than 2p(^r1), where p(n) denotes the partition function. By the asymptotics of the partition function of Hardy-Ramanujan (see [10]), p(t) ~ 4tT3exp where t = (q - 1)/2. So in particular, for any A G Awt0(GS) the probability that /A is nontrivial is less than p(t) (q) 1. Consequently, P(E) < (q2 - 1)(q2 - q)2^ ^ 0 as q Hence asymptotically almost surely, every S c Fq satisfies Aut0(GS) = {A/ : A G Fq}. The second statement follows trivially from this conclusion and this completes the proof of the lemma. □ Resuming the proof of the theorem, let S be a subset of Fq of size ^^ such that Aut(GS) = {A/ + b : A G F*, b G V(GS)}; such a choice for S exists by the preceding lemma. For such S, the distinguishing chromatic number of GS is greater than its chromatic number. Furthermore, since GS admits all translations in AG(2, q) as automorphisms it follows that it is vertex transitive. □ In fact, the graph GS satisfies xd (Gs) = x(Gs) + 1 as we shall see now. Theorem 4.3. Let S c Fq be a set of size such that Aut0(GS) = {A/ : A G Fq}. Then xd (Gs) = q + 1. Proof. For 1 = y G S, consider the coloring of GS described by the color classes {/^ : P G Fq}. Assign the color q +1 to only the vertex (0,0) G V(GS). This forms a q +1 coloring of GSwhich is obviously a proper coloring. To show that this is distinguishing, let ^ be a color fixing nontrivial automorphism of GS. By Theorem 4.2, ^ maps (x, y) to (ax + b1, ay + b2) for some a, b1, b2 G Fq. Since ^ fixes (0,0) we have b1 = b2 =0 and a =1. This implies ^ = a/ and hence it is not color fixing; indeed ^ maps (1,1) to (a, a) and (a, a) G /1. □ Our second result in this section describes a family of graphs with very large automorphism groups - much larger than exponential in |V(G)|, but for which xd(G) = x(G). As N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 101 was proven in [2], we already know that the Kneser graphs K(n, r) with r > 3 satisfy the same. However, one might also expect that in such cases, distinguishing proper colorings are perhaps rare, or at the very least, that there do exist minimal proper, non-distinguishing colorings of G. It turns out that even this is not true. Theorem 4.4. Let K (n, r) denote the complement of the Kneser graph, i.e., the vertices of K (n, r) correspond to r element subsets of [n] and two vertices are adjacent if and only if their intersection is non-empty. Then for n > 2r and r > 3 xd (K (n,r)) = x(K (n,r)). Moreover, every proper coloring of K (n, r) is in fact distinguishing. Proof. First, observe that since Aut(K(n, r)) ~ Sn for n > 2r, the full automorphism group of K(n, r ) is also Sn. Consider a proper coloring c of K(n, r) into color classes C1,C2,... ,Ct. Note that for any two vertices v1, v2 in the same color class, v1 n v2 = 0. If possible, let a e Sn be a non-trivial automorphism which fixes Cj for each i. Without loss of generality let a(1) = 2. Observe that for the vertex vi = (1,2,..., r), its color class has no other vertex containing 1 or 2, so a maps {1,2,,... ,r} to {1,2,... ,r}. Again, with the vertex v2 = {1, 3,... ,r+1}, which is in color class C2 = C1, a maps v2 into {2, a (3),..., a(r+ 1)} = v2, so a(v2) n v2 = 0 by assumption. However, since a(i) e {1, 2,... ,r} for each 3 < i < r this yields a contradiction. □ 5 Bipartite graphs with large xd (G) In this section we describe a family of bipartite graphs whose distinguishing chromatic number is greater than any integer k, where k > 4. As we described in the introduction, the sense of non-triviality of these examples arises from a couple of factors. Our examples contain several copies of Kr,s as induced subgraphs. That by itself does not imply that the distinguishing chromatic number is at least r + s but it is suggestive. What makes these families nontrivial is the fact that the distinguishing chromatic number of these graphs is in fact r + s + 1. Again, in order to describe these graphs, let q > 5 be a prime power, and let n := (P, L) be a Desarguesian projective plane of order q. As is customary, we denote by [r], the set {1,2 ...,r}. The graph which we denote LGq (g> Kr,s has vertex set V(LGq (g> Kr,s) = (P x [r]) U (L x [s]), and for p e P ,l G L, and (i,j) G [r] x [s] we have (p, i) adjacent to (l,j) if and only if p G l. Another way to describe this graph goes as follows. The weak product LGq x Kr,s is bipartite and consists of two isomorphic bipartite components. The graph LGq ( Kr,s is one of the connected components. For each point p there are r copies of p in the graph LGq ( Kr,s; we call the set {(p, i) |i G [r]} the fiber of p, and denoted it by F(p). Similarly we denote by F(l), the set F(l) = {(l, i) : i G [s]}, and shall call this the fiber of l. Each vertex (p, i) (resp. (l,j)) of LGq ( Kr,s has degree r(q + 1) (resp. s(q + 1)). Theorem 5.1. xd (LGq ( Kr,s) = r + s +1, where r,s > 2 and q > 5 is a prime number. Proof. Firstly, we show that xD(LGq ( Kr,s) > r + s. If possible, let C be an (r + s)-proper distinguishing coloring of LGq ( Kr,s and let Ci, i G [r + s] be the color classes of C in LGq ( Kr,s. We claim: 102 Ars Math. Contemp. 12 (2017) 111-126 1. For each p eP, each vertex of F (p) gets a distinct color. The same also holds for each l e L and each vertex of F(l). 2. If Cp and Cc denote the sets of colors on the vertices of |J F(p) and |J F(l) per lec respectively, then Cp n Cc = 0 and |Cp | = r, |Cc| = s. Consequently, for each i, either F (p) n C = 0 for each p eP or F (l) n C, = 0 for each l e L. We shall first prove each of the claims made above. 1. For p e P suppose Fcontains two elements, say (p, i) and (p,j), with the same color. Consider the map ^ that swaps (p, i) with (p,j) and fixes all other vertices. It is easy to see that ^ is a graph automorphism which fixes each color class C, contradicting the assumption that C is distinguishing. The argument for the part regarding vertices in the fiber F(l) is identical. 2. Let l e L and p e l. By claim 1 each vertex in F(p) has a distinct color. Since |F(p) | = r we may assume without loss of generality let (p, i) gets color i for i e [r]. In that case, no vertex of F(l) can be colored using any color in [r]. Furthermore, by the same reasoning as above, each vertex of F(l) is colored using a distinct color, so we may assume again that (l, i) is colored r + i for i = 1, 2..., s. Since there is a unique line through any two points, no vertex of the form (p',j) gets a color r+s r in U Cj. Similarly, no vertex of the form (l',j) belongs to U C,. Therefore, i=r+1 i=1 r all points and their fibers belongs to U C, and all lines with their fibers belongs to i=1 r+s u Cj. i=r+1 From claims 1 and 2 above, we conclude that for each p e P, C, n F (p) = 0 for i e [r]. Otherwise, since |F (p)| = r, there exist an i e [r] such that |C, n F (p)| > 2, contradicting claim 1. Similar arguments show that for each l e L, Cj+r n F (l) = 0 for i e [s]. To show C is not a distinguishing coloring we produce a nontrivial automorphism of LGq ( Kr,s which fixes each C, for i = 1,2,... ,r + s. We first set up some terminology. For i e [r], we call a vertex in the fiber of p its ith vertex if its color is i and shall denote it p1. Similarly, we shall call a vertex in the fiber of l its ith point if its color is i + r and shall denote it by l\ Let ^ e Aut(LGq) be a nontrivial automorphism such that ^(P) = P so that it also satisfies r^(L) = L. Let a be defined on V (LGq ( Krs) by a(v11) = v) for v ePuL. It is clear that a is a color preserving map. Moreover a preserves adjacency in LGq ( Kr s; indeed, v is adjacent to w in LGq if and only if F(v) U F(w) forms a Kr,s as a subgraph of LGq ( Kr,s and ^ e Aut(LGq). Therefore a is a nontrivial automorphism which fixes the color classes, thereby showing that xD(LGq ( Kr,s) > r + s. We now claim that xD (LGq ( Kr,s) < r + s + 1. For 1 < i < r - 1, assign the color i to the points {(p,i) : p e P} and for r +1 < j < r + s let {(l,j) : l e L} be colored j. Recall that LGq admits a distinguishing 3-coloring in which every vertex of L is given the same color, and the point set P is partitioned into P1, P2 that correspond to the other two color classes (Theorem 3.1). We split the set {(p, r) : p eP} into Cr := {(p, r)|p e P1} and Cr+su1 := {(p,r) : p e P2} and designate these sets as color classes r and r + s + 1 respectively. N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 103 It is easy to see that the above coloring is proper since adjacent vertices get different colors. To see that it is distinguishing, let ^ be a nontrivial automorphism which fixes each color class. Since ^ fixes each color class as a set, and ^ is nontrivial, in particular, ^ fixes the set {(p, r) : p € P}, and also fixes each set {(/, i) : / € L} for r + 1 < i < r + s, so in particular, ^ induces a nontrivial automorphism, v, on LGq — Cr U Ci+r for each i € [s], which is non-distinguishing. But this contradicts Theorem 3.1, and so we are through. □ 6 Concluding Remarks • It is possible to consider other Levi graphs arising out of other projective geometries (affine planes, incidence bipartite graphs of 1-dimensional subspaces versus k dimensional subspaces in an n dimensional vector space for some k etc). Many of our results and methods work in those contexts as well and it should be possible to prove similar results there as well, as long as the full automorphism group is not substantially larger. For instance, in the case of the incidence graphs of k sets versus /-sets of [n], it is widely believed (see [7], chapter 1) that in most cases, the full automorphism group of the generalized Johnson graphs is indeed Sn though it is not known with certainty. • As stated earlier, we believe that xd(LG4) — 3 though we haven't been able to show the same. Similarly, we believe xd (LG3) — 4. One can, by tedious arguments considering several cases, show that a monochromatic 3-coloring of LG3 is not a proper distinguishing coloring. For related details on what a monochromatic coloring is, see the Appendix. • We were able to show xd (Krxn) — r +1 since in this case, all proper r colorings of Krxn are of a specific type. For an arbitrary (prime) graph H, it is not immediately clear if xd(Hxn) > x(H). It would be interesting to find some characterization of graphs H with xd(Hxn) — x(H) + 1 for large n. • For a given k € N, we obtained nontrivial examples of family of vertex-transitive graphs G with arbitrarily large chromatic number which have xd(G) > x(G) and with |Aut(G)| somewhat small. It is an interesting question to seek infinite families of vertex-transitive graphs G with xd (G) > x(G) > k for any prefixed k, while |Aut(G)| — Ok (|G|). • While we have attempted to construct non-trivial families of bipartite graphs with large distinguishing chromatic number, it would be interesting to construct nontrivial examples of graphs with arbitrary chromatic number, and arbitrarily large distinguishing chromatic number. Acknowledgement We would like to thank the anonymous referees for helpful suggestions and comments that have helped in improving the clarity of the paper. References [1] M. O. Albertson and K. L. Collins, Symmetry breaking in graphs, Electron. J. 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Klavzar, Handbook of Product Graphs, Discrete Mathematics and its Applications, Taylor & Francis Group, LLC, Boca Raton, 2nd edition, 2011. [10] G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17 (1918), 75-115, doi:10.1112/plms/s2-17.1.75. [11] D. R. Hughes and F. C. Piper, Projective Planes, Graduate Texts in Mathematics 6, Springerverlag, New York, 1973. [12] W. Imrich, R. Kalinowski, F. Lehner and M. Pilániak, Endomorphism breaking in graphs, Electron. J. Combin. 21(1) (2014), #P 1.16. [13] C. Laflamme and K. Seyffarth, Distinguishing chromatic number of bipartite graphs, Electron. J. Combin. 16 (2009), #R76. [14] Rédei, Lacunary polynomials over finite fields, volume MR 50 # 4548, North-Holland publishing Co., Amsterdam-London; American Elsevier publishing Co., Inc., New York, 1973. [15] A. Russell and R. Sundaram, A note on the asympotics and computational complexity of graph distinguishability, Electron. J. Combin. 5 (1998), #R23. N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 105 7 Appendix 7.1 The Levi graph LG2 Firstly, we remark that the upper bound xd (G) < 2A - 2 whenever G is bipartite and G ^ ka-1,a, ka,a, which appears in [13], gives xd(LGq) < 2q. In particular, xd(LG2) < 4. We shall show that in fact xd (LG2) = 4. We first set up some notation, let {e^ e2, e3} be the standard basis of the vector space V with e1 = (1,0,0), e2 = (0,1,0) and e3 = (0,0,1). For g,h,k G Fq, a vector v G V is denoted by (g, h, k) if v = ge1 + he2 + ke3. A point p G P is denoted by (g, h, k) if p =< ge1 + he2 + ke3 > . Thus, there are q2 points in the form (1, h, k) such that h,k G Fq, q points in the form of (0,1, k) such that k G Fq and finally the point (0,0,1) to account for a total of q2 + q +1 points in PG(2, Fq). We start with the following definition. Definition 7.1. A coloring of the Levi graph is said to be monochromatic if all the vertices in one set of the vertex partition have the same color. Lemma 7.2. LG2 does not have a proper distinguishing monochromatic 3-coloring. Proof. Assume that LG2 has a proper distinguishing monochromatic 3-coloring. Without loss of generality let the line set L be colored with a single color, say red. Call the remaining two colors blue and green, say, which are the colors assigned to the vertices in P. We shall refer to the set of points that are assigned a particular color, say green, as the color class Green. By rank of a color class C (denoted r(C)), we mean the rank of the vector subspace generated by C. Observe that a nontrivial linear map T that fixes the color class Green, must necessarily also fix the color class Blue, so any such linear map would correspond to an automorphism that preserves each color class. For any 2-coloring of P (which has 7 points), one of the two color class has fewer than four points. Without loss of generality, assume that this is the color class Green. Firstly, if r (Green) < 2 then consider a basis B of V which contains a maximal linearly independent set of points in color class Green. If r(Green) = 2, then the linear map T obtained by swapping the elements of the color class Green in B, and fixing every other basis element is a non-trivial linear transformation of V which necessarily fixes the color class Green. If r(Green) = 1, then consider the map T which fixes the green point of B and swaps the other two (necessarily Blue) is a nontrivial linear transform that fixes the color class Green. Finally, if r(Green) = 3, then let T be the map that swaps two of them and fixes the third. Again, this map is a nontrivial linear map that fixes every color class. □ We now set up some notation. Denote the points in LG2 by e1, e2, e3, e1 + e2, e1 + e3, e2 + e3 and e1 + e2 + e3 (see Figure1) and denote the lines in the following way: 1. l1 : (e1 ,e2) the line (two dimensional subspace) spanned by e1 and e2. 2. I2 : (e1 ,e3). 3. I3 : (e2,e3). 4. I4 : (e1, e2 + e3). 5. I5 : (e2,e1 + e3). 6. l6 : (e3,e1 + e2). 106 Ars Math. Contemp. 12 (2017) 111-126 Figure 1: Fano plane 7. I7 : (ei + e3, e2 + e3). Theorem 7.3. xd (LG2) = 4. Proof. By the remark at the beginning of the section, we have xd (LG2) < 4, so it suffices to show xd(LG2) > 3. We first claim that if LG2 has a proper distinguishing 3-coloring, then three linearly independent points (points corresponding to three linearly independent vectors) get the same color. Suppose the claim is false. Then each monochrome set C of points satisfies r(C) < 2. Since any set of four points contains three linearly independent points and | V(LG2) | = 7, a 3-coloring yields a monochrome set of points of size exactly three. Denote this set by E and observe that E in fact corresponds to a line lE e L. Since any two lines intersect, no line is colored the same as the points of E. If p, p' eP \ E are colored differently, then the line lpp cannot be colored by any of the three colors contradicting the assumption. Consequently, every point in P \ E must be colored the same if the coloring were to be proper. But then this gives a color class with four points which contains three linearly independent points contradicting that the claim was false. Without loss of generality, suppose e1, e2, e3 are all colored red. Since l7 contains the points e1 + e2, e2 + e3 and e1 + e3, these three points cannot all have different colors. Hence at least two of these three points are in the same color class. Without loss of generality, assume that e1 + e2 and e2 + e3 have the same color. Now observe that the map a defined by a(e1) = e3,a(e3) = e1,a(e2) = e2, induces an automorphism of LG2 that fixes every color class within P. Furthermore a swaps l1 with l3 and l4 with l6 and fixes all the other lines. If the sets of lines {l4, l6 } and {11, l3} are both monochrome in L, then note that a fixes every color class contradicting that the coloring in question is distinguishing. Thus we consider the alternative, i.e., the possibilities that N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 107 the lines 11 and /3 (resp. /4 and /6) are in different color classes, and in each of those cases produce a non-trivial automorphism fixing every color class. Case I: /4 and /6 have different colors, say blue and green respectively. In this case, the point set witnesses at most two colors and none of the points of P \ {e1 + e3} can be colored blue or green. Moreover, by Lemma 7.2, all the seven points cannot be colored red (note that e1, e2, e3 are colored red). Consequently, e1 + e3 is colored, say blue, and all the other points are colored red. The /7, /5 and /2 are all colored green since all these three lines contain the point e1 + e3. As mentioned above, we shall in every case that may arise, describe a non-trivial automorphism a that fixes each color class. As before, we shall only describe its action on the set {e1, e2, e3}. Sub case 1 : /1 is colored blue. Then a(e1) = e1,a(e2) = e2 + e3,a(e3) = e3 fixes e1 + e3, swaps /1 with /4 and fixes /3. Consequently, it fixes every color class. Sub case 2 : /1 is colored green and /3 is colored blue. In this case, a(e1) = e2, a(e2) = e1,a(e3) = e1 + e2 + e3 doesthejob. Sub case 3: /1 and /3 are both colored green. In this case, the only line which is colored blue is /4. Then a(e1) = e2 + e3, a(e2) = e2, a(e3) = e1 + e2, does the job. From the above it follows that /4 and /6 cannot be in different color classes. So, we now consider the other possibility, namely that /1 and /3 are in different color classes. Case II: /6 and /4 have the same color but /1 and /3 are in different color classes, say blue and green respectively. Here we first note that e1 + e2 and e2 + e3 are necessarily red because they belong to /1 and /3 respectively. Again, we are led to three subcases: Sub case 1 : e1 + e3 and e1 + e2 + e3 are both colored blue. Here, it is a straightforward check to see that every / = /1 is colored green. Then, one can check that a(e1) = e1 + e2, a(e2) = e2, a(e3) = e3 fixes every color class. Sub case 2 : The point e1 + e3 is colored red and e1 + e2 + e3 is colored blue. Again, one can check in a straightforward manner, that for all 3 < i < 6, /i is colored green. If /2 is blue then a(e2) = e3,a(e3) = e2,a(e1) = e1 does the job. If /2 is colored green, a(e1) = e2, a(e2) = e1, a(e3) = e3 does the job. Sub case 3: e1+e2+e3 is colored red and e1+e3 is colored blue. Here we first observe that /2, /3, /5, /7 are all necessarily green. Also, by the underlying assumption (characterizing Case II), /4, /6 bear the same color. In this case, a(e1) = e1 + e2, a(e3) = e2 + e3, a(e2) = e2, does the job. This exhausts all the possibilities, and hence we are through. □ 7.2 The Levi graph LG3 As remarked earlier, it is not too hard to show that x(LGq) < 6, so the same holds for q = 3 as well. The next theorem shows an improvement on this result. Theorem 7.4. xd(LG3) < 5. Proof. As indicated earlier we denote the points p e P as mentioned in the beginning of this section. A line corresponding to the subspace {(x, y, z) e P : ax + by + cz = 0} is denoted (a, b, c). We color the graph using the colors 1,2,3,4, 5 as in Figure 2 (the color is indicated in a rectangular box corresponding to the vertex) It is straightforward to check that the coloring is proper. For an easy check we provide below, a table containing adjacencies of each p e P. Here the first row lists all the points in the projective plane of order 3. The column corresponding to the vertex p e P lists the set of lines / e L such that p e /, so that the 108 Ars Math. Contemp. 12 (2017) 111-126 Points ^ 100 110 010 120 112 121 012 122 011 111 101 102 001 Lines 001 001 001 001 011 011 011 012 012 012 010 010 010 ; 011 120 100 110 120 121 122 122 121 120 122 121 120 012 121 101 111 101 102 100 101 100 102 102 101 100 010 122 102 112 112 110 111 110 112 111 112 111 110 columns are the adjacency lists for the vertices in P. To see that this coloring is distinguishing, firstly, observe that the line 001 is the only vertex with color 1. Therefore, any automorphism ^ that fixes every color class necessarily fixes this line. Consequently, the points on 001 are mapped by ^ onto themselves. Since each point on 001 bears a different color, it follows that ^ fixes each p e 001. In particular, for 1 < i < 4, ^ maps each set [In, li2, li3} onto itself. Here, [lij, 1 < j < 3} denotes the set of lines adjacent to the ith point of 001. But again note that by the coloring indicated, the vertices lij and j have different colors for each i, so ) = lij for each pair (i, j) with 1 < i < 4,1 < j < 3. Now it is a straightforward check to see that ^ = I. □ 7.3 SAGE code to calculate f (G) when G = PGL(Fp SAGE has inbuilt functions that allow us to list the elements of PGL (F^) and to write down the disjoint cycle decomposition of a given permutation. For a e PGL(F^) the disjoint cycle decomposition, including its fixed points gives 6a, the total number of distinct orbits induced by a as in Lemma 2.2. Now it is easy to calculate P(Ba) and sum over PGL(F^). The SAGE code that we used for the calculation is given below, note that the N. Balachandran and S. Padinhatteeri: \d (G), \ Aut(G) |, and a variant of the Motion Lemma 109 text in the square bracket is an explanation of the corresponding line in the code. #!/usr/bin/env sage -python import sys from sage.all import * p = int(input("Enter p = " )) t = float(input("Enter t = " )) etot = p**2 + p + 1 [Total number of points in LGp.] y = 0 G = PGL(3,p) [Automorphism group of PG(3,p).] n = int(G.order()) [cardinality of G.] for i in range(1, n): g = G[i] s = Set(g.cycle_tuples(singletons=True)) [Set of disjoint cycles of g G G including singleton.] og = s.cardinalityO [Gives total number of distinct orbits induced by 'g'. That is 9g in Lemma 2.2.] ex = float(etot - og) [|C — 1| — 9g as in the proof of Lemma 2.2.] pg = 1 / float(t**ex) y = float(pg + y) print "tot prob is",y ¿^creative , ars mathematica ^commons contemporánea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 111-126 Coherent configurations over copies of association schemes of prime order Reza Sharafdini * Department of Mathematics, Faculty of Science, Persian Gulf University, Bushehr 75169-13817, Iran. Mitsugu Hirasaka Department of Mathematics, College of Natural Sciences, Pusan National University, Busan 609-735, South Korea. Received 21 November 2014, accepted 17 May 2016, published online 2 November 2016 Abstract Let G be a group acting faithfully and transitively on Qi for i = 1,2. A famous theorem by Burnside implies the following fact: If |Qi| = |Q2| is a prime and the rank of one of the actions is greater than two, then the actions are equivalent, or equivalently |(a,3)G| = |Qi| = |Q21 for some (a, ¡3) e Qi x Q2. In this paper we consider a combinatorial analogue to this fact through the theory of coherent configurations, and give some arithmetic sufficient conditions for a coherent configuration with two homogeneous components of prime order to be uniquely determined by one of the homogeneous components. Keywords: Coherent configurations, association schemes, prime order, symmetric designs. Math. Subj. Class.: 05C15, 05C10 1 Introduction A famous theorem by Burnside states that each transitive permutation group of prime degree with rank greater than two is Frobenius or regular. Since any Frobenius group of prime degree is a subgroup of one-dimensional affine group, it follows that such a permutation group is uniquely determined by its rank and degree up to equivalence of group actions. Especially, if a group acts faithfully, transitively but not 2-transitively on each of two sets of the same prime size, then the two actions are equivalent. Let us formulate this fact in the following two paragraphs. * Corresponding author. E-mail addresses: sharafdini@pgu.ac.ir (Reza Sharafdini), hirasaka@pusan.ac.kr (Mitsugu Hirasaka) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 112 Ars Math. Contemp. 12 (2017) 111-126 Let G be a group acting transitively on Q, for i = 1, 2. Then G acts on Q, x Qj by (a,|)g = (ag ,lg) for (a,|) G Q, x Qj and g G G, for all i,j = 1, 2. It is well-known that (e.g., see [6, Lemma 1.6B]) the following are equivalent: (a) The action of G on Q1 is equivalent to that on Q2; (b) There exists (a, |) G Q1 x Q2 such that Ga = Gp; (c) There exists (a, l ) G Qi x Q2 such that |(a,|)G| = |Qi| = |Q21. Note that the rank of the action of G on Q, is equal to the number of orbits of G on Q, x Q,, and if G acts faithfully on Q,, then G can be identified with a permutation group of Q,. Suppose that G acts faithfully on Q, with i = 1,2 and |Q11 = |Q21 is a prime. Then, as mentioned in the first paragraph, these actions are equivalent if the rank of one of the actions is greater than two, and so there exists an orbit R of G on Q1 x Q2 such that |R| = | Q11 = |Q21. In this paper we consider a combinatorial analogy to this fact through the theory of coherent configurations. The concept of coherent configurations was first introduced by Higman who published a series of papers (e.g., [11], [12], [13]) to associate a lot of important criterions with group actions. Here we define a coherent configuration, its intersection numbers and its fibers according to the notations as in [7]. Definition 1.1. Let V be a finite set and R a partition of V x V. We say that the pair C = (V, R) is a coherent configuration if it satisfies the following: 1. The diagonal relation AV is a union of elements of R where we denote {(u,u) | u g U} by Au for a set U. 2. For each R gR its transpose Rl = {(u, v) | (v, u) G R} is an element of R. 3. For all R,S,T gR there exists a constant cRS such that cRS = |R(u) n S4(v)| for all (u, v) G T, where we denote by T (w) the set {z g V | (w, z) G T} for w G V and T gR. The constants cRS are called the intersection numbers. A subset X of V is called a fiber of C if Ax G R. We denote the set of all fibers of C by Fib(C). By Definition 1.1(i), V is partitioned into the fibers of C, and by Definition 1.1(i),(iii), R is partitioned into \Rx,y | X,Y g Fib(C) j where Rx,y = [R G R | R C X x Let U be a union of fibers of C. Then the pair (U, {R G R | R C U x U^, is also a coherent configuration, which is denoted by CU. For R G Rx,y we denote c^t by dR. Then, by two-way counting we have |R| = d^X| = dRt ^|. (1.1) R. Sharafdini and M. Hirasaka: Coherent configurations over copies of association schemes . 113 For X G Fib(C), CX is nothing but an association scheme, i.e., a coherent configuration with only one fiber (see [2] or [20] for its background). For short we shall write RX,X as RX and CX is called a homogeneous component of C. A general question here is formulated as follows: what can be said about the coherent configuration if its homogeneous components are known. For example, it is a well-known fact that the coherent configuration corresponds to a system of linked block designs if |RX| = 2 for all X G Fib(C). After the seminal Hanaki-Uno theorem on association schemes of prime order (see [10] or Theorem 3.1), it seems quite natural to ask on a possible structure of a coherent configuration each homogeneous component of which is of prime order. The following is our first main result answering to this question: Theorem 1.2. Let X, Y G Fib(C) such that |X| = |Y| is a prime. Then |RX,Y| G {1, |RX |}. In particular, if |RX,Y | > 1, then |Rx,Y | = |RX | = |RY |. In order to state our second main theorem we need to recall the following observation. Let G be a group acting on a finite set Q. Then G acts on Oxfi componentwise, and an orbit of G on Q x Q is called an orbital (or 2-orbit) of G. We denote the set of orbitals of G by OG. Then it is well-known that CG = (Q, OG) is a coherent configuration, and Fib(CG) is the set of orbits of G on Q. In this sense, a coherent configuration is a combinatorial object to generalize the orbitals of a group action. Now we assume that C = ( V, R) is a coherent configuration with exactly two fibers X, Y. Then (1.1) proves the equivalence of the first two statements of the following (see [16] for the remaining): (d) There exists R G RX,Y such that |R| = |X| = |Y|. (e) 1 G {dR | R G Rx,y} n {dR | R G Ry,x}. (f) C is isomorphic to CX T2 where 7n = ({1,2,... ,n}, {{(i, j )} | 1 < i, j < n} (see Section 2 for the definition of isomorphism and 0). We notice the following: (d) is a combinatorial analogy to (c), and such R is a matching between X and Y ; (e) is a simple arithmetic condition on intersection numbers; (f) implies that CX and CY are isomorphic, and C is uniquely determined by CX. In this paper we aim to obtain the analogous conclusion (d)-(f) to (a)-(c). The following is our second main result to generalize the fact as in the first paragraph under certain arithmetic conditions on intersection numbers: Theorem 1.3. Suppose that C = ( V, R) is a coherent configuration with exactly two fibers X, Y satisfying |X| = |Y| isaprime, |Rx,x | > 2 and |Rx,y | > 1. (1.2) Then there exists R G RX,Y such that |R| = |X | = |Y | if one of the following conditions X1 holds with k = —--—- : |Rx,X | - 1 (i) |RX,X | > k2 (k + e — 2) where e is the number of prime divisors of k; 114 Ars Math. Contemp. 12 (2017) 111-126 (ii) k G {q, 2q, 3q} for some prime power q; (iii) k = 4q for some prime power q with 3 { q +1. Let us show the reason why we exclude the case of |RX,X | = 2. Each symmetric design induces the coherent configuration with exactly two fibers and eight relations (see [14] or [16, Example 1.3]), and if the design is a non-trivial one on a prime number of points, like the Fano plane, then the induced coherent configuration does not satisfy (d)-(f). Of course, if |Rx,y| — 1, then none of (d)-(f) hold, while C is the direct sum of Cx and CY (see [16] for the definition of direct sum). Remark 1.4. Applying Theorem 1.3 for CXUY with |X| < 100 we obtain the same conclusion as Theorem 1.3 except for the case (|X|,k) — (71, 35) (see Section 5 for the details). Suppose that (|X|, k) — (71,35) and 1 G {dr | R G Rx,y}. (1.3) Then by Theorem 1.2, |RX,Y | — 3. The three elements of RX,Y must form three symmetric designs whose parameters (v, k, A) are (71, 35,17), (71,21,6) and (71,15,3), respectively. Though each of such symmetric designs exists (see [1], [3], [5], [9] and [17] or [4, II.6.24,VI.16.30]), it does not guarantee the existence of a coherent configuration satisfying (1.3). In [14], Higman gave a result to eliminate the case of (|X|, k) — (71,35) as in the previous paragraph. But, the proof given in [14, (3.2)] contains a serious gap, so the result may not be recognized to be true, while we have not found any counterexample. We would be able to disprove [14, (3.2)] if there exists a coherent configuration satisfying (1.3). In Section 2 we prepare several basic results on intersection numbers and introduce the concepts of complex products and equitable partitions. In Section 3 we give a proof of Theorem 1.2. In Section 4 we give a proof of Theorem 1.3. We add Section 5 for the elimination of coherent configurations on at most 200 points satisfying (1.2). 2 Preliminaries Throughout this section we assume that C — (V, R) is a coherent configuration. Let Ci — (Vi, Rj) be a coherent configurations, i — 1, 2. An isomorphism from Ci to C2 is defined to be a bijection ^ : Vl URi —> V2 U R2 such that for all u,v G Vl and R G Rl, (u,v) G R ^ (V>(u), V(v)) G V(R). We say that Cl is isomorphic to C2 and denote it by Cl ~ C2 if there exists an isomorphism from Cl to C2. We set Rl R2 — {Rl < R2 | Rl g Rl, R2 G R2j, where Ri < R2 — \ ((ui, U2), (vi,v2^ | (ui, vi) G Ri, (u2,v2) G R2 k R. Sharafdini and M. Hirasaka: Coherent configurations over copies of association schemes . 115 Then (V1 x V2, R1 ® R2) is a coherent configuration called the tensor product of C1 and C2 and denoted by C1 (g) C2. Following [20] we define the complex product on the power set of R. For all subsets S and T of R we define the complex product ST of S and T to be the subset {r gR| 3(S,T ) GSxT ; cRT > o}. The complex product is an associative binary operation on the power set of R where the proof is parallel to that for association schemes (see [20]). For convenience we shall write S{T}, {S}T and {S}{T} as ST, ST and ST, respectively. In this paper we need intersection numbers cRS for R e Rx,y, S e RY,Z and T G RX,Z under the assumption |X| = |Y| = |Z|. The following is a collection of simplified equations on such intersection numbers (see [19] or [16, Lemma 2.2] for general formed equations1). For U C R we shall write du instead of ^veu dv. Lemma 2.1. For all X, Y, Z e Fib(C) with |X| = |Y| = |Z| and all R e Rx,y, S e Ry,z and T e RX,Z we have the following: 1. dRds = cRsdr; T eUx,z 2. cRSdT = cRSt dR = cRtTds and lcm(dR, ds) | cRSdT; 3. |{U e R | cRs > 0}| < gcd(dR, ds), i.e., |RS| < gcd(dfl, ds); 4. |X| = dRx,X = dRx.y. The following lemmata were proved in [18, Lemma 2.3, Lemma 2.2]2: Lemma 2.2. For all S,T e RX,Y with |X| = |Y|, we have SSt n tt 1 C {Ax } if and only if c§tT < 1 for each R e R. Lemma 2.3. Let Z e Fib(C) such that |Z | is a prime. Then for each R e RZ \ {AZ} we have: Z1 1. dR = k where k = • |Rz |- 1' 2. seRZ css4 = k — 1. According to [8] or [15] we define an equitable partition of a homogeneous component. Definition 2.4. Let X e Fib(C) and n = {C1, C2,..., Cm} be a partition of X, i.e., m X = J Ci, Ci n Cj = 0 if i = j, and C, = 0 for each i = 1, 2,..., m. i= i An element of n is called a cell. We say that n is an equitable partition of CX if, for all i, j = 1, 2,... ,m and each R e RX, |R(x) n Cj | is constant whenever x e C,. 1 We missed to assume that all fibers of c have the same size at Lemma 2.2 in [16] where the lemma is used only for such coherent configurations in [16]. 2Though it is a statement for association schemes, a parallel way to the proof can be applied for balanced coherent configurations. 116 Ars Math. Contemp. 12 (2017) 111-126 For example, {X} and {{x} | x G X} are equitable partitions of CX. For each Y G Fib(C) and each y G Y we define ny := {T(y) | T G Ry,x }. (2.1) Then ny is an equitable partition of CX, since |R(x) n S(y)| = cRSt whenever x G T(y). 3 Proof of Theorem 1.2 In [10] Hanaki and Uno proved the following brilliant theorem: Theorem 3.1. All non-principal irreducible characters of an association scheme of prime order are algebraic conjugate and of degree one. The following proposition is obtained as a consequence of the previous theorem: Proposition 3.2. Let C = (V, R) be an association scheme of prime order and n be an equitable partition of C. Then |n| = 1 mod |R| — 1. Proof. Let A denote the adjacency algebra of C over C. Then the subspace W spanned by the characteristic vectors of the cells in n is a left A-module with respect to the ordinary matrix product. Since A is semi-simple, W is a direct sum of irreducible submodules. Note that the subspace spanned by the all-one vector is an A-submodule of W affording the principal character, and its multiplicity is one. Since the character afforded by W is integral valued, it is left invariant from any algebraic conjugate action. It follows from Theorem 3.1 that all non-principal irreducible submodules of W have the same multiplicity, say m. Since dime(W) = |n| and dimc(A) = |R|, it follows that |n| = 1 + m(|R| — 1). □ Proof of Theorem 1.2. Let C = (V, R) be a coherent configuration with X, Y G Fib(C) such that |X | = | Y | is a prime. Recall that ny is an equitable partition of CX where y G Y. By (2.1), |ny | = |Rx,y |. Then it follows from Proposition 3.2 that |Rx,y | = 1 mod |Rx | — 1. Since |Rx,y| < |Rx| (see [13, p.223] or [16, Proposition 2.7]), |Rx,y| G {1, |Rx|}. Applying the first statement for CY with |RX,Y | < |RY |, we obtain the second statement. □ R. Sharafdini and M. Hirasaka: Coherent configurations over copies of association schemes . 117 4 Proof of Theorem 1.3 For the remainder of this paper we assume that C = (V, R) is a coherent configuration with X, Y e Fib(C) such that m = |X| = |Y| is a prime, r = |RX| > 2 and |Rx,y| > 1. By Theorem 1.2, we have r = |Rx | = |Rx,y | = |Ry |. For the remainder of this paper we set m — 1 k = -. r — 1 By Lemma 2.3(i) the multi-set (dfl | R e Rz ) with Z e {X, Y} coincides with (1, k,..., k) by a suitable ordering. In this section we aim to show that 1 e {dR | R e RX,Y}, which implies that the multi-set (dR | R e RX,Y) coincides with (1, k,..., k) by a suitable ordering, since the complex product SR is a singleton with dSR = ds whenever S e RX and dR = 1 by Lemma 2.1(iii). Lemma 4.1. For all S, T e RX,Y with S = T we have the following: (i) ds ds = ds mod k; (ii) ds dT = 0 mod k. Proof. (i) Applying Lemma 2.1(i) for S and S4 with ds = dst and c^Jft = ds, we obtain that ds ds = ds + k ^ cSSt. T eRx,x T= Ax (ii) Applying Lemma 2.1(i) for S and T4 with dT = dTt and AX e ST4, we obtain that ds dT = k cST t. T eRv □ We set Si := {T e Rx,y | k t dT}, S2 := {T e Rx,y | dT = k} and S3 := {T e Rx,y | k | dT, k < dT}. Lemma 4.2. Let k = p^1 • • • Pae where p4 are the distinct prime divisors of k and a are positive integers. Then we have the following: 1. For each i = 1,..., e there exists a unique S e RX,Y such that pi t ds; 2. |S11 < e; 3. kjSsj + ds1 < 1 + k(e — 1). 118 Ars Math. Contemp. 12 (2017) 111-126 Proof. (i) By Lemma 2.1(iv) and Lemma 2.3(i), m = 1 + (r — 1)k = 1 mod p». Since m = dRx,Y, there exists an S G RX,Y such that pj \ ds. The uniqueness of such S is a direct consequence of Lemma 4.1(ii). (ii) The correspondence given in (i) gives a function from jpi,p2,... ,pe} to Si. It remains to show that this function is onto. Let S G S1. By the definition of S1, there exists p» such that p^ does not divide ds. By Lemma 4.1(i), dsds = ds mod k. Therefore ds(ds — 1) is divided by k. Since ds and ds — 1 are relatively prime, p"i \ ds implies that p» \ ds .It follows from (i) that ds lies in the range of the function. (iii) Note that r = |Si | + S | + |S? | and 3 m = ^ ds = ^dSi > d5l + k|S2| + 2k|S31. seRx.Y ¿=1 Since k|S21 + k|S3| = k(r — |Si |) and m = 1 + k(r — 1), it follows that 1 + k(|Si| — 1) > dsi + k|Ss|. By (ii), we have 1 + k(e — 1) > ds1 + k|Ss|. This completes the proof of (iii). □ Lemma 4.3. We have max{ds | S G RX,Y} < k • min{ds | S G RX,Y}. Proof: Let S, T G RX,Y such that ds = min{ds | S G RX,Y} and dT := max{ds | S G RX,Y}. Then T G RS for some R G RX since T G RX S. Applying Lemma 2.1(i) we have dT < kds. □ For S g RX,Y we define Us := {R G RX | R4Rn SS1 = {AX}}. Lemma 4.4. For each S G RX,Y we have the following: 1. r — |US| < (ds — 1)(k — 1). 2. If R G Us — {AX }, then k divides dT for each T G RS. 3. If Us S n S2 = 0, then r < ds (k + e — 2). Proof. (i) Note that rx —Us = (J {R G rX | Ri G R4R}. Riess'-{Ax } R. Sharafdini and M. Hirasaka: Coherent configurations over copies of association schemes . 119 By Lemma 2.1(iii) with c^X > 0, ISS4 -{Ax }| < ds - 1. It follows from Lemma 2.3(ii) that |{R e Rx I Ri € R4R}| < ^ cR1r = k - 1. ReR This implies that r -|Us| = |Rx -Us|< (ds - 1)(k - 1). (ii) It is an immediate consequence of Lemma 2.1(ii) and Lemma 2.2. (iii) Suppose that Us s ns2 = 0. Then we have UsS C Rx,y - S2. It follows from (ii) that (Us -{Ax })S C S3. By Lemma 4.2(iii) and Lemma 4.3, dsa < dsk|S3| < ds[1 + k(e - 1) - d5l]. (4.1) On the other hand, applying Lemma 2.2 and Lemma 2.1(iv) for the first inequality and (i) for the second one, duss > 1 + (|Us| - 1)k - ds > 1 + [r - (ds - 1)(k - 1) - 1]k. (4.2) Since (Us - {Ax})S C S3, dUss - ds < d(Us-{Ax})s < dS3 . It follows from (4.1) and (4.2) that 1 + [r - (ds - 1)(k - 1) - 1]k - ds < ds[1 + k(e - 1) - d^], and hence, Thus, r < dS[2 + k(e - 1) - dSl] - k + (ds - 1)(k - 1) + 1. r < ds[k2 + e - 1 - d|1 + k - 1] - k + 2 - k < ds(k + e - 2). This completes the proof of (iii). □ Proposition 4.5. If r > k2 (k + e — 2) where e is the number of prime divisors of k, then 1 e {ds | s g }. 120 Ars Math. Contemp. 12 (2017) 111-126 Proof. We claim that min{ds | S e RX,Y} < k. If not, then 1 + k(r — 1) = m ^^ ds > kr, seUx,v a contradiction. By Lemma 4.3, max{ds | S e RX,Y} < k2. Applying the contraposition of Lemma 4.4(iii) we have Us S n S2 = 0 for each S e Rx,y , and hence, T e RS for some R e Us and T e S2. Since dT = k and cRs = 1 by Lemma 2.2, ds divides k for each S e RX,Y. This implies that |S3| = 0. We claim |S41 = 1. Suppose not. Since 1 + (r — 1)k = m = dSl + k(r — |Si |), 1 + k|S1| < k + ^ ds < k + k/2 + k/2+ (|S1| — 2)k, sesi a contradiction. By the claim we have S1 = {S} for some S e RX,Y. Since 1 + k(r — 1) = m = k|S2| + ds = k(r — 1) + ds, we have ds = 1. This completes the proof. □ Lemma 4.6. If S, T e Rx,y with ST4 = {R}, then cRr > dT for each R1 e SS4 and cRr > ds for each R2 e TT Proof. Let y e Y, x1,x2 e S4(y) and z e T4(y). Note that (xj,z) e R for i = 1,2 since ST4 = {R}. Since z e T4(y) is arbitrarily taken, we have T4(y) Ç R(x1) n R(x2), which proves the first statement. By the symmetric argument the second statement can be proved. □ Proposition 4.7. There exist no S, T e RX,Y such that ST4 = {R}, ds + dT > k + 1 and 1 < ds < dT. (4.3) Proof. Suppose that S, T e RX,Y satisfies (4.3). We claim that SS4 = {AX,R1} for some R1 e RX — {AX}. Suppose not, i.e., SS4 — {AX } has at least two elements R1, R2. By Lemma 2.1(i), k2 = dfldflt > k + cRRtdfll + cRRdfl2 = k + k + k. It follows from Lemma 4.6 and ds + dT > k +1 that k2 > k(k + 2), a contradiction. We claim that SS4 n TT4 = {Ax, R1}. Suppose not, i.e., SS4 n TT4 = {Ax}. Then, by Lemma 2.2, cRTt = 1. It follows from Lemma 2.1(i) that k = dR = dsdT, which contradicts ds + dT > k +1 and 1 < ds < dT. R. Sharafdini and M. Hirasaka: Coherent configurations over copies of association schemes . 121 We claim that R = R4. Suppose not, i.e., R = R4. Then, by Lemma 2.3(ii), cR1 > cR1 I cR1 > da cr2r2 - cRflt + cRtfl - dS k - != E cR1r2 - CRRt + ^R - dS + dT - k + 1, a contradiction. We claim that TT4 = {AX,RJ. If R2 G TT4 - {AX,RJ, then cRR - ds by Lemma 4.6 with R = R4. By Lemma 2.1(i), k2 = dfldR - k + cRRk + cRRk, which implies that k - 1 + dT + dS, a contradiction to dS + dT - k +1. We claim that cR^ - dT - 2. By the previous claim, for all zi, z2 G Twith zi = z2 we have (z1, z2) G R1. Thus, cRU* = |Ri(zi) n Ri(z2)| - |T4(y) - {zi, Z2}| - dT - 2. Since Cr1 Rt + cRRt - dT - 2 + dT - k by Lemma 4.6, it follows from Lemma 2.3(ii) that R = Ri. Thus, cRRt = k - 1 since 1 < dS and S4(y) U T4(y) \ {xi,x2} Ç R(xi) n R(x2) for xi,x2 G S4(y). Since {AX, R} is closed under the complex product, 1 + k divides |X|. Since |X| is a prime, it follows that {AX, R} = , and hence |RX | = 2, a contradiction. □ Lemma 4.8. Suppose that k = 4q for some prime power q and 1 G {ds | S G }. Then S1 =0, |Si| = 2, and {ds | S G Si} = {3q, q + 1}. Proof. By Lemma 4.2(iii) and the assumption, |S3| = 0. By Lemma 4.2(ii), |Si | < 2. Let S G Si. Then, by Lemma 4.1, dS = 1 mod q. By the assumption, 1 < dS < 4q. Since ds < dS1 < 1 + 4q Lemma 4.2(iii), it follows from Lemma 4.1 that ds G {q + 1, 3q + 1}. Let T G with S = T. Since dSdT = 0 mod 4q by Lemma 4.1, q | dT. Since m = 1 + k(r - 1) = dS1 + dS2 = dS + dT + k(r - 2), we have dS + dT = k + 1. Therefore, we conclude from Proposition 4.7 that {dS | S G Si} = {3q, q + 1}. □ Proof of Theorem 1.3. (i) is a direct consequence of Proposition 4.5. (ii) Suppose on the contrary that 1 G {ds | S gRx,y}. Note that e < 2 if k G {q, 2q, 3q} for some prime power q. By Lemma 4.2(iii), |S3| = 0, and dS1 < k + 1. Since 1 + k(r - 1) = ds1 + ds2 < k + 1 + ds2, we have dS2 - k(r - 2), and, hence, |S2| - r - 2. Suppose k = q. Then the statement follows from Lemma 4.2(iii) since e =1. 122 Ars Math. Contemp. 12 (2017) 111-126 Suppose k = 2q. Then |Si| < 2 and {ds | S G Si} = {q, q +1} by Lemma 4.2(ii),(iii) and Lemma 4.1. Without loss of generality we assume that Si = {S, T}, ds = q + 1 and dT = q. Since q and q +1 are relatively prime, it follows from Lemma 2.1(iii) that ST4 = {R} for some R gR, which contradicts Proposition 4.7. Suppose k = 3q. Then we have either {ds | S G Si} = {q, 2q + 1} or {ds | S G Si} = {2q, q + 1}. The first case is done by Proposition 4.7. For the last case we assume that Si = {S, T}, ds = q +1 and dT = 2q. By Lemma 2.1(i),(ii), SS4 = {AX,R} for some R G R with R = R4. This implies that k = dR is even since |X | is an odd prime, so q is a power of two. Thus, ds and dT are relatively prime. Therefore, the statement follows from Lemma 2.1(iii) and Proposition 4.7. (iii) Suppose k = 4q. Then, by Lemma 4.8, {ds | S G RX,Y} = {q, 3q + 1} or {ds | S G RX,Y} = {3q, q + 1}. The statement follows from the assumption and Proposition 4.7. □ 5 Appendix In this section we show how Theorem 1.3 is applied to small configurations CXUY with |X| = |Y| < 100. First, we denote by M the set of primes m less than 100. Second, we take the set K of positive integers k such that k | m — 1 for some m G M with k < m — 1 and k G {q, 2q, 3q | q is a prime power} U {4q | q is a prime power with 3 \ q +1}. Then K = {20,30, 35,44}. Lemma 5.1. If k = 20, then 1 G {ds | S G Rx,y}. Proof. Suppose not. By Lemma 4.8, {ds | S G Si} = {15,6}. Let S G Rx,y with ds = 6. By Lemma 2.1(ii), 6 | cfstk for R G SS1 \ {AX}. Thus, 3 | cfst, which contradicts Lemma 2.1(ii). □ Lemma 5.2. Suppose that each element of RY = {AY, R, R'} is symmetric and nx = {C1, C2, C3 } is the equitable partition of (Y, RY) as in Section 2 for x G X .We define {ßij}i i. When n = 3, those graphs are exactly Tower of Hanoi graphs. Later, those graphs have been called Sierpinski graphs in [7] and they were studied by now from numerous points of view. The reader is invited to read, for instance, the following recent papers [2, 5, 4, 7, 8, 9] and references therein. This construction was generalized in [3] for any graph G, by defining the t-th generalized Sierpinski graph of G, denoted by S(G, t), as the graph with vertex set V4(G) and edge set defined as follows. {u, v} is an edge if and only if there exists i e {1,..., t} such that: (i) uj = vj, if j < i; (ii) uj = vj and {uj, vj} e E(G); (iii) uj = vj and vj = uj if j > i. Figure 1: A graph G and the generalized Sierpinski graph S(G, 2) Figure 1 shows a graph G and the Sierpinski graph S(G, 2), while Figure 2 shows the Sierpinski graph S(G, 3). Notice that if {u, v} is an edge of S(G, t), there is an edge {x, y} of G and a word w such that u = wxyy ... y and v = wyxx ... x. In general, S(G, t) can be constructed recursively from G with the following process: S(G, 1) = G and, for t > 2, we copy n times S (G, t - 1) and add the letter x at the beginning of each label of the vertices belonging to the copy of S(G, t - 1) corresponding to x. Then for every edge {x, y} of 130 Ars Math. Contemp. 12 (2017) 111-126 G, add an edge between vertex xyy... y and vertex yxx ... x. See, for instance, Figure 2. Vertices of the form xx ... x are called extreme vertices. Notice that for any graph G of order n and any integer t > 2, S(G, t) has n extreme vertices and, if x has degree d(x) in G, then the extreme vertex xx ... x of S(G, t) also has degree d(x). Moreover, the degrees of two vertices yxx... x and xyy ... y, which connect two copies of S(G, t - 1), are equal to d(x) + 1 and d(y) + 1, respectively. 1. To the best of our knowledge, [14] is the first published paper studying the generalized Sierpiriski graphs. In that article, the authors obtained closed formulae for the Randic index of polymeric networks modelled by generalized Sierpiriski graphs. In this note we consider the case where every internal vertex of G is a cut vertex and we obtain a closed formula for the strong metric dimension of S(G, t). 3 The strong metric dimension of S(G, t) The following basic lemma will become an important tool to prove our main results. Lemma 3.1. Let G be a connected graph. If v is a cut vertex of G, then v ^ d(G). Proof. Let v G V (G) be a cut vertex and x G V (G)-{v}. Let G1 be the connected compo- J. A. Rodrigues-Velazquez andE. Estaji: Strong metric dimension of Sierpinski graphs 131 nentof G -{ v } containing x andlet G2 bea connected component of G—{ v } different from Gi. Since there exists y G V(G2) which is adjacent to v in G and dG(x, v) < dG(x, y), we conclude that x and v are not mutually maximally distant in G. □ An end-vertex is a vertex of a graph that has exactly one edge incident to it, while a support vertex is a vertex adjacent to an end-vertex. Theorem 3.2. Let G be a connected graph and let -(G) be the number of end-vertices of G. Then, dims (G) > -(G) - 1. Moreover, if every vertex of degree greater than one is a cut vertex, then the bound is achieved. Proof. Let G be a connected graph. Since the set Q(G) of end-vertices of G is a subset of d(G) and the subgraph of GSR induced by Q(G) is a clique, we conclude that a(GSR) > -(G) - 1. Hence, by Theorem 1.1 we obtain the lower bound. Now, if every vertex of degree greater than one is a cut vertex, by Lemma 3.1 we have that d(G) is equal to the set of end-vertices of G. Then GSR = K|e(G)| and so Theorem 1.1 leads to dims(G) = -(G) - 1. □ From now on, we will say that a vertex of degree greater than one in a graph G is an internal vertex of G. We shall show that if every internal vertex of G is a cut vertex, then the bound above is achieved for S(G, t). To begin with, we state the following lemma. Lemma 3.3. Let G be a graph of order n having -(G) end-vertices. For any positive integer t, the number of end-vertices of S(G, t) is , , u -(G) (ri - 2nt-i + 1) -(S(G, t)) = ---L. n - 1 Proof. In this proof, we denote by Sup(G) the set of support vertices of G. Also, if x G Sup(G), then -G(x) will denote the number of end-vertices of G which are adjacent to x. Let t > 2. For any x G V(G), we denote by Sx(G,t - 1) the copy of S(G,t - 1) corresponding to x in S(G, t), i.e., Sx(G, t - 1) is the subgraph of S(G, t) induced by the set {xw : w G Vt-1(G)}, which is isomorphic to S(G, t - 1). To obtain the result, we only need to determine the contribution of Sx(G, t - 1) to the number of end-vertices of S(G, t), for all x G V(G). By definition of S(G, t), there exists an edge of S(G, t) connecting the vertex xy ... y of Sx(G, t - 1) with the vertex yx ... x of Sy (G, t - 1) if and only if x and y are adjacent in G. Hence, an end-vertex xy ... y of Sx(S(G, t - 1) is adjacent in S(G, t) to a vertex yx... x of Sy (G, t - 1) if and only if y is an end-vertex of G and x is its support vertex. Thus, if x G Sup(G), then the contribution of Sx(G, t -1) to the number of end-vertices of S(G, t) is -(S(G, t - 1)) - -G(x) and, if x G Sup(G), then the contribution of Sx(G, t - 1) to the number of end-vertices of S(G, t) is -(S(G, t - 1)). Then we obtain, -(S(G,t)) = (n -| Sup(G)|)-(S(G,t - 1)) + ^ (-(S(G,t - 1)) - -g(x)) xGSup(g) = n-(S(G, t - 1)) - -(G). 132 Ars Math. Contemp. 12 (2017) 111-126 Now, since -(S(G, 1)) = -(G), we have that (nt-1 - 1)' -(S(G, t)) = -(G) (ni-1 - nt-2-----n - 1) = -(G) ( nt-1 - n - 1 J Therefore, the result follows. □ The following result is a direct consequence of Theorem 3.2 and Lemma 3.3. Theorem 3.4. Let G be a connected graph of order n having -(G) end-vertices and let t be a positive integer. Then , , u e(G) (n4 - 2nt-1 + 1) - n +1 dims(S(G,t)) > --->--. n - 1 As we will show in Theorem 3.6, the bound above is tight. Lemma 3.5. Let G be a connected graph and let t be a positive integer. If every internal vertex of G is a cut vertex, then every internal vertex of S(G, t) is a cut vertex. Proof. As above, for any x G V(G), we denote by Sx(G, t - 1) the copy of S(G, t -1) corresponding to x in S(G, t). We proceed by induction on t. Let S(G, 1) = G be a connected graph such that every internal vertex is a cut vertex and assume that every internal vertex of S(G, t - 1) is a cut vertex. We differentiate two cases for any internal vertex xw of S(G, t), where x G V(G) and w G Vt-1(G). Case 1. w has degree one in S(G, t - 1). In this case xw has degree two in S(G, t). Hence, xw is adjacent to x1 w', for some x1 G V (G)-{x}, and then w = x1 x1... x1, w' = xx ... x, x1 is an end-vertex of G and x is the support of x1. As a result, {xw, x1w'} is the only edge connecting vertices in Sx1 (G, t - 1) to vertices outside the subgraph SX1 (G, t - 1). Therefore, xw is a cut vertex of S(G, t). Case 2. w is a cut vertex of S(G, t - 1). In this case, we take two connected components C1 and C2 obtained by removing w from S(G, t - 1). Suppose, for contradiction purposes, that xw is not a cut vertex of S(G, t). Then there exist two neighbours x1 , xk of x and a sequence of subgraphs SX1 (G, t - 1), Sx2 (G, t -1),..., SXk (G, t - 1) such that x1... x1 G V(C1), xk ... xk G V(C2) and there exists an edge of S(G, t) connecting Sxi (G, t - 1) to Sxi+1 (G, t - 1), for all i G {1, 2,..., k}. Note that the only vertices connecting Sxi (G, t - 1) and Sxi+1 (G, t -1) are xjxi+1xi+1... xj+1 and xi+1xjxj... x4, where xj and xj+1 are adjacent in G. Hence, x, x1, x2,..., xk, x is a cycle in G, and so there is a cycle in S(G, t - 1) of the form Pxxi ,Pxix2,^^2x3,... ,Pxfc_ixfc, Pxfcx, where Px^+i is the path of order 2t-1 from xjxH ... x4 to xi+1 xi+1... xi+1 composed by binary words on alphabet } (the paths Pxx1 and Pxkx are defined by analogy) and we identify the vertex xjxj... xj of two consecutive paths PXi-lXi and PXiXi+1 to form the cycle. As a result, there are two disjoint paths from x1x1... x1 to xkxk.... xk, which contradicts the fact that x1x1... x1 G V(C1) and xkxk.... xk G C2. Therefore, xw is a cut vertex of S(G, t). According to the two cases above, we conclude the proof by induction. □ J. A. Rodrigues-Velazquez andE. Estaji: Strong metric dimension of Sierpinski graphs 133 Our next result is obtained from Theorem 3.2 and Lemma 3.5. Theorem 3.6. Let G be a connected graph of order n having -(G) end-vertices and let t be a positive integer. If every internal vertex of G is a cut vertex, then , , u e(G) (n/' - 2ni-1 + l) - n +1 dims(S(G,t)) = ----. n-1 Obviously, if the base graph is a tree, then we can apply the formula above. In particular, we would emphasize the following particular case of this result, where K1r denotes the star graph of r leaves and Pr denotes the path graph of order r. Corollary 3.7. For any integers r,t > 2, • dims(S(Ki,r,t)) = (r +1)t-1(r - 1). • dims(S(Pr,t))=2rt - ^ r + 3. r-1 Let G be a graph of order n and let H = {H1,H2,... ,Hn} be a family of graphs. The corona product graph G © H is defined as the graph obtained from G and H by taking one copy of G and joining by an edge each vertex of Hi with the iih-vertex of G. These graphs were defined by Frucht and Harary in [1]. Corollary 3.8. Let G be a graph of order n and let H = {H1, H2,..., Hn} be a family of empty graphs of order ni, respectively. 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ARS MATHEMATICA CONTEMPORANEA 12 (2017) 135-144 Remarks on the thickness of Kn,n,n Yan Yang * Department of Mathematics, Tianjin University, Tianjin, P.R.China Received 10 March 2015, accepted 14 March 2016, published online 3 November 2016 Abstract The thickness 6(G) of a graph G is the minimum number of planar subgraphs into which G can be decomposed. In this paper, we provide a new upper bound for the thickness of the complete tripartite graphs Kn n n (n > 3) and obtain 6(Kn n n) = , when n = 3 (mod 6). Keywords: Thickness, complete tripartite graph, planar subgraphs decomposition. Math. Subj. Class.: 05C10 1 Introduction The thickness 6(G) of a graph G is the minimum number of planar subgraphs into which G can be decomposed. It was defined by Tutte [11] in 1963, derived from early work on biplanar graphs [2, 10]. It is a classical topological invariant of a graph and also has many applications to VLSI design, graph drawing, etc. Determining the thickness of a graph is NP-hard [7], so the results about thickness are few. The only types of graphs whose thicknesses have been determined are complete graphs [1,3], complete bipartite graphs [4] and hypercubes [5]. The reader is referred to [6, 8] for more background on the thickness problems. In this paper, we study the thickness of complete tripartite graphs Kn,n,n, (n > 3). When n = 1,2, it is easy to see that K1,1,1 and KC2,2,2 are planar graphs, so the thickness of both ones is one. Poranen proved 6(Kn,n,n) < [f] in [9] which was the only result about the thickness of Kn,n,n, as far as the author knows. We will give a new upper bound for 6(Kn,n,n) and provide the exact number for the thickness of Kn,n,n, when n is congruent to 3 mod 6, the main results of this paper are the following theorems. Theorem 1.1. For n > 3, 6(Kn,n,n) < [^2 + 1. Theorem 1.2. 6(K„,„,„) = [^2 when n = 3 (mod 6). * Supported by the National Natural Science Foundation of China under Grant No. 11401430 E-mail address: yanyang@tju.edu.cn (Yan Yang) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 136 Ars Math. Contemp. 12 (2017) 111-126 2 The proofs of the theorems In [4], Beineke, Harary and Moon determined the thickness of complete bipartite graph Km,n for almost all values of m and n. Lemma 2.1. [4] The thickness of Km,n is |~2(m+w-2)l except possibly when m and n are odd, m < n and there exists an integer k satisfying n = L2^^—"]. Lemma 2.2. For n > 3, 0(Kn,n,n) > []. Proof. Since K„,2„ is a subgraph of K„,„,„, we have 0(Kn,n,n) > 6(Kn,2„). From Lemma 2.1, the thickness of Kn,2n (n > 3) is [n^1], so the lemma follows. □ For the complete tripartite graph Kn,n,n with the vertex partition (A, B, C), where A = {ao,..., an-1}, B = {bo,..., bn-i} and C = {c0,..., Cn-i}, we define a type of graphs, they are planar spanning subgraphs of Kn n n, denoted by G[ aibj+iCk+i], in which 0 < i,j,k < n - 1 and all subscripts are taken modulo n. The graph G[ aibj+iCk+i] consists of n triangles aibj+iCk+i for 0 < i < n - 1 and six paths of length n - 1, they are aobj+iCfc+2a3bj+4Cfc+5 . .. asibj+si+iCk+3i+2 . .., cka1bj+2Ck+3a4bj+5 . . . Ck+3ia3i+1bj+3i+2 . . . , bjCk+ia2bj+3Ck+4a5 . .. bj+3iCk+3i+ia3i+2 . .., aoCk+ibj+2a3Ck+4bj+5 . .. a3iCk+3i+ibj+3i+2 . .., bjaiCk+2bj+3a4Ck+5 . .. bj+3ia3i+iCk+3i+2 . .., Ckbj+ia2Ck+3bj+4a5 . . . Ck+3ibj+3i+ia3i+2 . . . . Equivalently, the graph G[aibj+iCk+i] is the graph with the same vertex set as Kn,n,n and edge set {aibj+i-i, aibj+i, aibj+i+i, aiCk+i-i, aiCk+i, aiCk+i+i | 1 < i < n - 2} U{bj+iCk+i-i, bj+iCk+i, bj+iCk+i+i | 1 < i < n - 2} U{aobj, aobj+i, an-ibj+n-2, an-ibj+n-i} U{aoCk ,aoCk+i, an-iCk+n-2 , ar—iCk+n-i)} U{bj Ck ,bj Ck + i,bj+n-iCk+n-2,bj+n-iCk+n-i}. Figure 1(a) illustrates the planar spanning subgraph G[aibiCi] of K5 5 5. Y. Yang: Remarks on the thickness of K„,„,„ 137 Figure 1: A planar subgraphs decomposition of K5j5j5 Theorem 2.3. When n = 3p + 2 (p is a positive integer), #(Kni„in) < p + 2. Proof. When n = 3p + 2 (p is a positive integer), we will construct two different planar subgraphs decompositions of Kn n n according to p is odd or even, in which the number of planar subgraphs is p + 2 in both cases. Case 1. p is odd. Let Gi,..., Gp be p planar subgraphs of K„,„in where Gt = G[aibi+3(i-1)ci+6(i-1)L for 1 < t < ; and Gi = — 1)ci+6(t—1)+2], for ^i3 < t < p and p > 3. From the structure of G[ajbj+ick+i], we get that no two edges in G1,..., Gp are repeated. Because subscripts in Gt, 1 < t < p are taken modulo n, {3(t - 1) (mod n) | 1 < t < p} = {0, 3,6,..., 3(p - 1)}, {6(t - l) (modn) | 1 < t < pt-1 } = {0,6,..., 3(p -1)} and {6(t - 1) + 2 (modn) | < t < p} = {3, 9,..., 3(p -2)}, the subscript sets of b and c in Gt, 1 < t < p are the same, i.e., 138 Ars Math. Contemp. 12 (2017) 111-126 {i + 3(t - 1) (mod n) | 1 < t < p} = {i + 6(t - 1) (mod n) | 1 < t < ^y1} U {i + 6(t - 1) + 2 (modn) | < t < p}. Furthermore, if there exists t G {1,... ,p} such that ajbj is an edge in Gt, then ajCj is an edge in Gk for some k G {1,... ,p}. If the edge ajbj is not in any Gt, then neither is the edge ajCj in any Gt, for 1 < t < p. From the construction of Gt, the edges that belong to Kn n n but not to any Gt, 1 < t < p, are aob3(t-i)-i, aoC3(t_i)_i, 1 < t < p (1) an-1&3(t-1), an_lC3(i-1), 1 < t < p (2) aA-3, ajbi-2, 0 < i < n - 1 (3) ajCj-3, ajCj-2, 0 < i < n - 1 (4) p + 3 bjCj+3(t-i)-i, bjCi+3(i-i), 0 < i < n - 1 and t = —— (5) b3(t-1)C6(t-1)-1, b3(t-i)-iC6(t-i), 1 < t < (6) p + 3 ^(i-i^i-i^^ b3(i-1)-1C6(i-1) + 2, —< t < p andp > 3 (7) Let Gp+ i be the graph whose edge set consists of the edges in (3) and (5), and Gp+2 be the graph whose edge set consists of the edges in (1), (2), (4), (6) and (7). In the following, we will describe plane drawings of Gp+1 and Gp+2. (a) A planar embedding of Gp+1. Place vertices b0, b1,..., bn-1 on a circle, place vertices ai+3 and cj+ n+i in the middle of b and bj+1, join each of aj+3 and ci+ n+i to both bj and bj+1, we get a planar embedding of Gp+1. For example, when p = 1, n = 5, Figure 1(b) shows the subgraph G2 of K5j5j5. (b) A planar embedding of Gp+2. Firstly, we place vertices c0 , c1 ,..., cn-1 on a circle, join vertex ai+3 to Cj and ci+1, for 0 < i < n - 1 , so that we get a cycle of length 2n. Secondly, join vertex an-1 to C3(t-1) for 1 < t < p, with lines inside of the cycle. Let ^ be the line drawn inside the cycle joining an-1 with C6(t-1)-1 if 1 < t < or with C6(t-1)+1 if < t < p (p > 3). For 1 < t < p, insert the vertex b3(t-1) in the line Thirdly, join vertex a0 to C3(t-1)-1 for 1 < t < p, with lines outside of the cycle. Let 4 be the line drawn outside the cycle joining ao with C6(t-1) if 1 < t < P+1 or with C6(i-1)+2 if P+3 < t < p (p > 3). For 1 < t < p, insert the vertex b3(t-1)-1 in the line In this way, we can get a planar embedding of Gp+2. For example, when p = 1, n = 5, Figure 1(c) shows the subgraph G3 of K5,5,5. Summarizing, when p is an odd positive integer and n = 3p+2, we get a decomposition of Kn n n into p + 2 planar subgraphs G1,..., Gp+2. Case 2. p is even. Let G1,..., Gp be p planar subgraphs of K„,„in where Gt = G[ajbj+3(i-1)Ci+6(i-1)+3],for 1 < t < p;and Gt = G[aibj+3(t-1)Ci+6(i-1) + 2],for < t < p. With a similar argument to the proof of Case 1, we can get that the subscript sets of b and c in Gt, 1 < t < p are the same, i.e., {i + 3(t - 1) (modn) | 1 < t < p} = {i + 6(t - 1) + 3 (modn) | 1 < t < p} U {i + 6(t - 1) +2 (modn) | ^y2- < t < p}. Y. Yang: Remarks on the thickness of K„,„,„ 139 From the construction of Gt, Gp and G p+2 have n - 2 edges in common, they are bi+3(p+2_1)Ci+6(p+2_!)+1, 1 < i < n - 1 and i = n - 4, we can delete them in one of thes2e two graphs2 to avoid repetition. The edges that belong to K„in,m but not to any Gt, 1 < t < p, are a0b3(t_1)_1, a0C3(i_1)_1? 1 < t < P (8) a„_i63(t_i), a„_1C3(t_1), 1 < t < p (9) aj6j_3, aj6j_2, 0 < i < n — 1 aiCj_3, ajCj_2, 0 < i < n — 1 bjCj_1, bjCj, bjCi+1, 0 < i < n - 1 p b3(i_1)C6i_4, 1 < t < 2 P + 2 »3(i_1)C6i_5, —^ 2. Then, we show that BHn is a lexicographic product of an n-valent graph Xn and the null graph with two vertices, where Xn is a Zn-1 -regular cover of the n-dimensional hypercube Qn. Keywords: Automorphism group, balanced hypercube, Cayley graph, arc-transitive. Math. Subj. Class.: 05C25, 20B25 1 Introduction The hypercube is widely known as one of the most popular interconnection networks for parallel computing systems. As a variant of the hypercube, the balanced hypercube was proposed by Huang and Wu [8] to enhance some properties of the hypercube. An n-dimensional balanced hypercube, denoted by BHn, is defined as follows. Definition 1.1. For n > 1, BHn has 4n vertices, and each vertex has a unique n-component vector on {0,1,2,3} for an address, also called an n-bit string. A vertex (a0, a\,..., an_i) is connected to the following 2n vertices: ( ((ao + 1)(mod 4), ai,..., aj_i, a*, ai+i,..., an_i), \ ((ao - 1)(mod 4), ai,..., a_i,ai, ai+i,..., an_i), *This work was supported by the National Natural Science Foundation of China (11271012, 11571035, 11231008) and the Fundamental Research Funds for the Central Universities (2015JBM110). t Corresponding author. E-mail addresses: jxzhou@bjtu.edu.cn (Jin-Xin Zhou), jinkwak@postech.ac.kr (Jin Ho Kwak), yqfeng@bjtu.edu.cn (Yan-Quan Feng), 10271079@bjtu.edu.cn (Zhen-Lin Wu) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 146 Ars Math. Contemp. 12 (2017) 145-154 f ((ao + 1)(mod 4), ai,..., a4_i, (ai + (-1)°° )(mod 4), ai+i,..., an_i), \ ((ao - 1)(mod 4),ai,.. .,ai_i, (ai + (-1)°° )(mod 4),am,.. .,a„_i), for 1 < i < n — 1. (0, 0) (3, 0) (0, 3) (3, 3) (1, 1) (2, 1) (1, 2) (2, 2) Figure 1: Two balanced hypercubes: BH1 and BH2 By now, various properties of the balanced hypercube, such as, Hamiltonian laceabil-ity, bipanconnectivity, super connectivity etc. have been extensively investigated in the literature [7, 8, 9, 14, 16, 17, 18, 19]. In many situations, it is highly desired to use interconnection networks which are highly symmetric. This often simplifies the computational and routing algorithms. It has been shown that the balanced hypercube is vertex-transitive and arc-transitive (see [14, 22]). When dealing with the symmetry of graphs, the goal is to gain as much information as possible about the structure of the full automorphism groups. Recently, several publications have been put into investigation of automorphism groups of Cayley graphs having connection with interconnection networks (see, for example, [5, 10, 20, 21]). In [22], it was proved that BHn is an arc-transitive Cayley graph. Definition 1.2. For n > 1, let Hn be an abelian group defined as follows: Hn (y) x (zi) x (z2) x ... x (z„_i) = Z2 x Z4 x Z4 x ... x Z4. The generalized dihedral group of Hn, denoted by Dih(Hn), is the semi-direct product of Hn by a group (x) of order 2 with the involution x inverting every element in Hn. Let Gn = Dih(Hn) = Hn x (x) and let S = {x, xy, xzi, xyzi | i = 1, 2,..., n — 1}. Let rn be the following Cayley graph: r„ = Cay(G„,S). Proposition 1.3. [22, Theorem 3.7] For each n > 1, BHn = rn is arc-transitive. Definition 1.4. Let Ln be a subgroup of Hn defined by (1.1) Ln (zi) x (z2) x ... x (zn_i) = Z4 x Z4 x ... x Z4. i 1 3 2 J.-X. Zhou et al.: Automorphism group of the balanced hypercube 147 Let Tn = Dih(Ln) = Ln x (x). Clearly, Tn is a subgroup of Gn of index 2. Set Q = = 1, 2,..., n — 1}, and define Xn as the following Cayley graph: Xn = Cay(Tn, Q). (1.2) For convenience, in what follows we shall always let rn = BHn. In [3], the authors proved the following result. Proposition 1.5. [3, Theorem 3.4] For each n > 1, BHn = Xn [2Ki], where Xn is defined as following: By Proposition 2.1, it is easy to see that Aut(BHn) = Z21 Aut(Xn) *. So, to determine the automorphism group of BHn, the key is to determine the automorphism group of Xn. In this paper, we prove that Xn is a 2-arc-transitive normal Cayley graph, and Aut(Xn) = R(Tn) X Aut(Tn, Q) = Tn X Sn. As the automorphism group of BHn is known, one may ask: Does BHn have a stronger symmetry property? In this paper, we show that BHn is arc-transitive but not 2-arc-transitive. As another application, we prove that Xn is a Zn-1-regular cover of the hypercube Qn. This, together with the fact BHn = Xn[2K1], gives a theoretical explanation of the relationship between BHn and Qn. 2 Preliminaries In this section, we shall introduce some notations, terminology and preliminary results. Throughout this paper only undirected simple connected graphs without loops and multiple edges are considered. Unless stated otherwise, we follow Bondy and Murty [2] for terminology and definitions. Let n be a positive integer. Denote by Zn the cyclic group of order n, by Sn the symmetric group of degree n and by Kn n the complete bipartite graph of order 2n and valency n, respectively. We also use nKi, Kn and Cn to denote the null graph, the complete graph and the cycle with n vertices, respectively. In a parallel computing system, processors are connected based on a specific interconnection network. An interconnection network is usually represented by a graph in which vertices represent processors and edges represent links between processors. Let G be a simple undirected connected graph. We denote by Aut(G) the full automorphism group of G, and by V(G) and E(G) the sets of vertices and edges of G, respectively. For u, v e V(G), denote by {u, v} the edge incident to u and v in G. For a vertex v in a graph G, use NG (v) to denote the neighborhood of v, that is, the set of vertices adjacent to v. *One can also obtain this by using [4, Theorem 5.7]. We thank a referee for pointing out this. 148 Ars Math. Contemp. 12 (2017) 111-126 An s-arc in a graph G is an ordered (s + 1)-tuple (v0, vi,..., vs_i, vs) of vertices of G such that vi_1 is adjacent to vi for 1 < i < s and vi_1 = vi+1 for 1 < i < s - 1. A graph G is said to be s-arc-transitive if Aut(G) is transitive on the set of s-arcs in G. In particular, 0-arc-transitive means vertex-transitive, and 1-arc-transitive means arc-transitive or symmetric. A graph G is edge-transitive if Aut(G) acts transitively on E(G). Clearly, every arc-transitive graph is both edge-transitive and vertex-transitive. 2.1 Wreath products of groups For a set V and a group G with identity element 1, an action of G on V is a mapping V x G ^ V, (v, g) ^ vg, such that v1 = v and (vg)h = vgh for v e V and g, h e G. The kernel of G acting on V is the subgroup of G fixing V pointwise. For two groups K, H, if H acts on K (as a set) such that (xy)h = xhyh for any x, y e K and h e H, then H is said to act on K as a group. In this case, we use K x H to denote the semi-direct product of K by H with respect to the action. Let H be a permutation group on a finite set A. For convenience, let A = {1,2, • • • , n}. Let G be a permutation group on a finite set and let N = G x G x ••• x G. v-v-' n times We define the action of H on N as following: Vh e H, (g1, g2 • • • , gn)h = (g^-1, g2h-i, • • • , gnh-1), gi e G, i = 1, 2, • • • , n. Then the semi-direct product of N by H with respect to this action is called the wreath product of G and H, denoted by G 2 H. Clearly, G 2 H = {(g1, g2, • •• ,gn; h) | gi e G, h e H}. Moreover, G 2 H can be viewed as a permutation group on $ x A as following: (x, i) (g1 'Bn,h) = (xgi ,ih). Let G and H be two graphs. The lexicographic product G[H] is defined as the graph with vertex set V(G) x V(H) and for any two vertices (u1 ,v1), (u2,v2) e V(G) x V(H), they are adjacent in G[H] if and only if either u1 = u2 and v1 is adjacent to v2 in H, or u1 is adjacent to u2 in G. In view of [13, Theorem.], we have the following. Proposition 2.1. [13, Theorem.] Let X and Y betwographs. Then Aut(X [Y]) = Aut(Y)2 Aut(X) if and only if (1) if there are two distinct vertices u, v e V (X) such that NX (u) = NX (v), then Y is connected; (2) if there are two distinct vertices u, v e V (X) suchthat NX (u)U{u} = NX (v)U{v}, then the complement Y of Y is connected. 2.2 Cayley graphs Let G be a permutation group on a set Q and a e 0. Denote by Ga the stabilizer of a in G, that is, the subgroup of G fixing the point a. We say that G is semiregular on Q if J.-X. Zhou et al.: Automorphism group of the balanced hypercube 149 Ga = 1 for every a e Q and regular if G is transitive and semiregular. Given a finite group G and an inverse closed subset S C G \ {1}, the Cayley graph Cay(G, S) on G with respect to S is defined to have vertex set G and edge set {{g, sg} | g e G, s e S}. A Cayley graph Cay(G, S) is connected if and only if S generates G. Given a g e G, define the permutation R(g) on G by x ^ xg, x e G. Then R(G) = {R(g) | g e G}, called the right regular representation of G, is a permutation group isomorphic to G. It is well-known that R(G) < Aut(Cay(G, S)). So, Cay(G, S) is vertex-transitive. In general, a vertex-transitive graph X is isomorphic to a Cayley graph on a group G if and only if its automorphism group has a subgroup isomorphic to G, acting regularly on the vertex set of X (see [1, Lemma 16.3]). For two inverse closed subsets S and T of a group G not containing the identity 1, if there is an a e Aut(G) such that Sa = T then S and T are said to be equivalent, denoted by S = T. The following proposition is easy to obtain. Proposition 2.2. If S and T are equivalent then Cay(G, S) = Cay(G, T). A Cayley graph Cay(G, S) is said to be normal if R(G) is normal in Aut(Cay(G, S)) (see [15]). Let Cay(G, S) be a Cayley graph on a group G with respect to a subset S of G. Set A = Aut(Cay(G, S)) and Aut(G, S) = {a e Aut(G) | Sa = S}. Proposition 2.3. [15, Proposition 1.5] The Cayley graph Cay(G, S) is normal if and only if Ai = Aut(G, S), where Ai is the stabilizer of the identity 1 of G in A. 2.3 Covers of graphs An important tool in studying symmetry properties of graphs is the covering technique. An epimorphism p : X ^ X of graphs is called a regular covering projection if there is a semiregular subgroup CT(p) of the automorphism group Aut(X) of X whose orbits in V (X) coincide with the vertex fibers p-i(v), v e V (X), and the arc and edge orbits of CT(p) coincide with the arc fibers p-i((u, v)), u ~ v, and the edge fibers p-i({u, v}), u ~ v, respectively. In particular, we call the graph XX a regular cover of the graph X. The semiregular group CT(p) is the covering transformation group. If CT(p) is isomorphic to an abstract group N then we speak of XX as a regular N-cover of X. For more results on the covering of graphs, we refer the reader to [6, 12]. Let X be a connected k-valent graph and let G < Aut(X) act transitively on the 2-arcs of X. Let N be a normal subgroup of G. The quotient graph XN of X relative to N is defined as the graph with vertices the orbits of N in V(X) and with two orbits adjacent if there is an edge in X between those two orbits. In view of [11, Theorem 9], we have the following. Proposition 2.4. If N has more than two orbits in V(X), then N is semiregular on V(X), XN is a k-valent graph with G/N as a 2-arc-transitive group of automorphisms, and X is a regular N-cover of XN. 3 Automorphism group of the balanced hypercube In this section, we shall determine the full automorphism group of the balanced hypercube. From Proposition 1.5 we know that rn = Xn[2Ki], and by Proposition 2.1, Aut(T„ ) = Z21 Aut(Xn). So, the key step is to determine the automorphism group of Xn. 150 Ars Math. Contemp. 12 (2017) 111-126 Lemma 3.1. For n > 1, Xn is a 2-arc-transitive normal Cayley graph, and furthermore, Aut(Xn) = R(Tn) x Aut(Tn, 0), where R(Tn) = Tn = Dih(Zn—1) and Aut(Tn, 0) = Sn. Proof. Clearly, X1 = K2 and X2 = C8. It is easy to see that the statement is true for these two cases. In what follows, assume that n > 3. We first prove the following two claims. Claim 1 Aut(Tn, 0) = Sn. Since 0 generates Tn, Aut(Tn, 0) acts faithfully on 0, and hence Aut(Tn, 0) < Sn. It is easy to verify that xz1, z-1zi(2 < i < n - 1), z—1 generate Tn and they satisfy the same relations as x, zj(1 < i < n - 2), zn—1. This implies that the map a : x ^ xz1, zi ^ z-1zi+1 (1 < i < n — 2), zn_1 ^ z—1, induces an automorphism of Tn. Clearly, for each 1 < i < n — 2, (xzj)a = xzi+1, and x ^ xz1 and (xzn—1)a = x. This implies that a cyclicly permutates the elements in 0, and so a G Aut(Tn, 0). Similarly, for each 2 < i < n — 1, we define a map as the following: Pi : x ^ x, z1 ^ zj, zj ^ z1, zj ^ zj(1 < i,j < n — 1 and i = j). Then pi induces an automorphism of Tn, and furthermore, Pi interchanges xz1 and xzi and fixes all other elements in 0. Hence, G Aut(Tn, 0) and by elementary group theory, we obtain that the subgroup generated by Pi(2 < i < n — 1) is isomorphic to Sn—1. Since Sn—1 is maximal in Sn, one has (a, p | 2 < i < n — 1} = Sn. It follows that Aut(Tn, 0) = (a, | 2 < i < n — 1} = Sn. Claim 2 For any xzi, there are (n — 2) 6-cycles in Xn passing through the 2-arc (x, 1, xzi), namely, Cj'j = (1, x, z—1, xziz—1, z—1zi, xzi, 1) with j = i and 1 < j < n — 1. By Claim 1, Aut(Tn, 0) acts 2-transitively on 0. It is well-known that a vertex-transitive graph is 2-arc-transitive if and only if the vertex-stabilizer Aut(Xn)v is 2-transitive on the set of vertices adjacent to v. So, Xn is 2-arc-transitive. To prove the claim, it suffices to show that the statement is true for the case when i = 1. First, for any 2 < j < n — 1, one may easily check that C1j = (1, x, z_ 1, xz1z— 1, z1zj— 1, xz1,1) is a 6-cycle passing through the 2-arc (x, 1,xz1). Let C' be an arbitrary 6-cycle passing through (x, 1,xz1). Then there exist s1,s2,t1,t2 G 0 such that C' = (1, x, s1x, s2«1x = ¿2^1xz1, t1xz1, xz1,1), where S1 = x, s2 = s1, ¿1 = xz1 and ¿1 = ¿2. Clearly, s1 = xzj for some 1 < j < n — 1. In the rest of the proof of Claim 2 the following well-known fact will be frequently used. Fact Every element in (z1} x (z2} x ... x (zn _ 1} can be uniquely written in the following form z?1 z?2 ...z'nnL11 ,ai G Z4(1 < i < n — 1). If s2 = x, then xxzjx = t2t1xz1. It follows that zjx = t2t1xz1 and hence zjz1 = t2t1. If t2 = x, then t1 = xzk for some 1 < k < n — 1, and so zjz1 = zk. By Fact, this is impossible. If t2 = xz« for some 1 < i < n — 1, then we have either t1 = x or t1 = xzp for some 1 < p < n — 1. For the former, we have zj z1 = z_1, and for the latter, we have t2t1 = xz«xzp = z—1zp = zjz1. From the above Fact, both of these cannot happen. J.-X. Zhou et al.: Automorphism group of the balanced hypercube 151 If s2 = xzj for some 1 < i < n — 1, then xz^xzjx = t^xz^ It follows that z—1 zjx = t2t1xz1 and hence z— 1zj-z1 = t211. If t1 = xzk and t2 = xzp for some 1 < k,p < n — 1, then t2t1 = z—1zk = z— 1zj-z1. This is also impossible. If t1 = x and t2 = xzp for some 1 < p < n—1,then t2t1 = z—1 = z— 1zj-z1. This is also impossible. So, we must have t1 = xzk and t2 = x for some 1 < k < n — 1. Then t2t1 = zk = z— 1zj-z1. Clearly, s1 = s2. Then zk = zj and zj = z1. That is s2 = xz1, t2 = x, t1 = s1 = xzj. It follows that C' = C 1'j = (1,x, z— 1,xz1z— 1,z— 1z1,xz1,1). 1 Figure 3: 6-cycles passing through (x, 1, xzj) Now we are ready to complete the proof. Let A = Aut(Xn) and let A1 be the stabilizer of the identity 1 in A. Let A^ be the kernel of A1 acting on Q. Then A^ fixes every element in Q. For any xzj (1 < i < n — 1), by Claim 2, there are exactly (n — 2) 6-cycles in Xn passing through the 2-arc (x, 1, xzj), namely, Ci,j = (1, x, z— 1, xzjz— 1, z— 1zj, xzj, 1) with j = i and 1 < j < n — 1 (see Fig. (3)). Note that the neighborhood of x is {1, z—1 11 < i < n — 1} and the neighborhood of xzj is {1, zj, z— 1zj | 1 < i, j < n — 1, j = i}. Since there are no 6-cycles passing through z—1, x, 1, xzj and zj, it follows that A1 fixes z—1 and zj (1 < i < n — 1) . By [3, Lemma 4.2], Xn has girth 6, and so Cj j is the unique 6-cycle passing through z— 1, x, 1, xzj, z— 1zj. As A| fixes z— 1, x, 1 andxzj, A| mustfix z— 1zj. By the arbitrariness of i, j, we obtain that A| fixes every vertex of the set {z— 1,zj,z— 1zj | 1 < i,j < n—1, j = i} which is just the set of vertices at distance 2 from the identity 1. By the vertex-transitivity and connectivity of Xn, A| fixes all vertices of Xn. It follows that A1 = 1, and so A1 acts faithfully on Q. Therefore, A1 < Sn. By Claim 1, Aut(Tn, Q) = Sn, and since Aut(Tn, Q) < A1, one has Aut(Tn, Q) = A1. By Proposition 2.3, Xn is normal, and so A = R(Tn) x Aut(Tn, Q). □ Now we are ready to determine the automorphism group of BHn. Theorem 3.2. For n > 1, Aut(BHn) = Z21 (Tn x Sn). Proof. By Proposition 1.5, BHn ^ Xn[2Kij. By Proposition 2.1, Aut(BHn) = Z2 \ Aut(Xn). From Theorem 3.1 we obtain that Aut(Xn) = R(Tn) x Aut(Tn, = Tn x Sn. It follows that Aut(BHn) = Z2 I (Tn x Sn). □ 152 Ars Math. Contemp. 12 (2017) 111-126 4 Related results As the automorphism group of BHn is known, we can obtain more information about the symmetry properties of BHn. By Proposition 1.3, BHn is arc-transitive, and by Theorem 3.1, Xn is 2-arc-transitive. It is natural to ask: whether BHn has much stronger symmetry property? We answer this in negative by showing that BHn is not 2-arc-transitive. Theorem 4.1. For n > 2, BHn is arc-transitive but not 2-arc-transitive. Proof. Suppose, by way of contradiction, that BHn is 2-arc-transitive. Recall that BHn = Cay(Gn, S). Then the vertex-stabilizer Aut(BH„)i of the identity 1 of Gn in Aut(BHn) is 2-transitive on S. That is, for any two distinct ordered pairs from S x S, say (u1, v1) and (u2,v2), there exists a e Aut(BHn)1 such that (u1,v1)a = (u2,v2). In particular, there exists a e Aut(BHn)1 such that (x, xy)a = (x, xz1). This implies that x and xz1 have the same neighborhood because x and xy have the same neighborhood. However, from [22, Lemma 3.8], we see that xy is the unique vertex which has the same neighborhood as x, a contradiction. □ By Proposition 1.5, BHn = Xn[2K1]. As a consequence of Theorem 3.1, we can also prove that Xn is a Z^-1-regular cover of the hypercube Qn. This reveals the relationship between the balanced hypercube BHn and the hypercube Lemma 4.2. For n > 1, let N = Zf. Let G = Cay(N, S) be a connected n-valent Cayley graph. Then G is isomorphic to the n-dimensional hypercube Qn. Proof. It is well-known that Qn is a Cayley graph on N with respect to the subset T = {(1,0,0, ••• , 0), (0,1,0, ••• , 0), ••• , (0,0,0, ••• , 1)}. Viewing N as an n-dimensional vector space on the field Z2, one may see that T is a basis of N. Since G is an n-valent Cayley graph, one has |S | = n, and since G is connected, one has N = (S). This means that S is also a basis of N. So, there is an automorphism of N which maps S to T. By Proposition 2.2, G = Qn, as desired. □ Theorem 4.3. For n > 3, Xn is a Zf—1 -regular cover of Qn. Proof. By Theorem 3.1, R(Tn) is normal in Aut(Xn). Remember that Tn = Dih(Ln) = Ln x (x), where Ln = (z1) x ... x (zn-1) = Z4 x ... x Z4, and x is an involution inverting every element in Ln. Set Z = (R(z2)} x ... x (R(zn_1)). t,—1 times is an involution inveiung every elemeutiu Ln. set z — (R^^ ^ /i?/^2 Then Z = Z2 x ... x Z2, v-^-' n— 1 times and Z is just the center of R(Tn). It follows that Z is characteristic in R(Tn). Since R(Tn) < Aut(Xn), one has Z < Aut(Xn). Consider the quotient graph Yn of Xn relative to Z. Clearly, Z is semiregular on the vertex-set of Xn, and so it has more than 2 orbits on V(X). Since Xn is 2-arc-transitive, by Proposition 2.4, Yn is also an n-valent graph with J.-X. Zhou et al.: Automorphism group of the balanced hypercube 153 Aut(Xn)/Z as a 2-arc-transitive automorphism group, and Xn is a Z-regular cover of Yn. To complete the proof, it suffices to prove that Yn = Qn. Noting that Z < R(Tn) and R(Tn) is regular on V(Xn), R(Tn)/Z is regular on V(Yn). It follows that Yn is a Cayley graph on R(Tn)/Z. As R(Tn) = Dih(Ln), one has R(Tn)/Z = Zn. Since Yn has valency n, by Lemma 4.2, one has Yn = Qn. □ Conclusion In [14], the authors introduced the balanced hypercube to enhance some properties of the hypercube. Graph symmetry is an important factor in the design of an interconnection network. In 1997, it has been shown that the balanced hypercube is vertex-transitive. Recently, it was shown that the balanced hypercube is also arc-transitive. However, the full automorphism group of the balanced hypercube remained unknown. In this paper, we solve this problem. As applications, we first analyze the symmetry properties of the balanced hypercube and show that the balanced hypercube is not 2-arc-transitive. Then, we give a theoretical explanation of the relationship between the balanced hypercube and the hypercube. Acknowledgements: The authors are indebted to the anonymous referees for many valuable comments and constructive suggestions. References [1] N. Biggs, Algebraic graph theory, Cambridge university press, 1993. [2] J. A. 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Yuan, Symmetric property and reliability of balanced hypercube, IEEE Transactions on Computers 64 (2015), 876-881. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 155-165 Uniquely colorable Cayley graphs Walter Klotz Institut fur Mathematik, Technische Universität Clausthal 38678 Clausthal-Zellerfeld, Germany Torsten Sander Fakultat Informatik, Ostfalia Hochschule für angewandte Wissenschaften 38302 Wolfenbuttel, Germany Received 24 June 2015, accepted 5 January 2016, published online 15 November 2016 It is shown that the chromatic number x(G) = k of a uniquely colorable Cayley graph G over a group r is a divisor of |r | = n. Each color class in a k-coloring of G is a coset of a subgroup of order n/k of r. Moreover, it is proved that (k - 1)n is a sharp lower bound for the number of edges of a uniquely k-colorable, noncomplete Cayley graph over an abelian group of order n. Finally, we present constructions of uniquely colorable Cayley graphs by graph products. Keywords: Vertex coloring, color classes, Cayley graph. Math. Subj. Class.: 05C15, 05C25 1 Introduction A proper k-coloring of an undirected graph G = (V, E) with vertex set V = V(G) and edge set E = E (G) is a map f : V ^ C from V into a set C with |C | = k elements ('colors') such that any two adjacent vertices are assigned different colors. If not otherwise stated a k-coloring is always understood to be a proper k-coloring. A graph G is k-colorable if it admits a k-coloring. The chromatic number x(G) is the smallest number k for which G is k-colorable. An optimal coloring of G is a x(G)-coloring of G. The color class of the coloring f : V ^ C with respect to color c G C consists of all vertices x G V with f (x) = c. If f is a k-coloring of G, then the color classes of f constitute a partition of V into at most k disjoint stable sets which means that any two elements of these sets are nonadjacent. The graph G is uniquely colorable if every optimal coloring of G induces the E-mail addresses: klotz@math.tu-clausthal.de (Walter Klotz), t.sander@ostfalia.de (Torsten Sander) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract 156 Ars Math. Contemp. 12 (2017) 111-126 same partition into color classes. If G is uniquely colorable, then we mean by the color classes of G the color classes of an optimal coloring of G. Let us point out some previous work on uniquely colorable graphs. Harary et al. [11] construct new ones from given uniquely colorable graphs. Bollobas [4] presents a lower bound for the minimal degree ¿(G) which forces G to be uniquely colorable. Hillar and Windfeldt [13] give an algebraic characterization of uniquely k-colorable graphs, which partly originates in ideas of Lovasz [16] and Bayer [3]. They also apply commutative algebra to develop an algorithm for recognizing unique colorability. Xu [19] establishes a sharp lower bound for the number of edges of a uniquely k-colorable graph on n vertices: |E| > (k - 1)n - Q . (1.1) Daneshgar [7] and Daneshgar, Naserasr [8] concentrate on cliques in uniquely colorable graphs. Special classes of uniquely colorable graphs are investigated by Akbari et al. [1], Chao and Chen [5], Chartrand and Geller [6]. The Cayley graph G = Cay(r, S) over the finite (multiplicative) group r with shift set (or symbol) S C r has vertex set V = V(G) = r and edge set E = E(G) = {{x,y} : x,y G r,xy-1 G S}. To avoid loops we demand that the unit element e G r is not in S. To make G undirected we require that S is self-inverse, S-1 = S, which means that s G S always implies s-1 G S. For general properties of Cayley graphs we refer to Godsil and Royle [9]. Circulant graphs are Cayley graphs over cyclic groups. We represent the cyclic group of order n by the additive group Zn of integers modulo n, Zn = {0,1,..., n - 1}. A well-known circulant graph is the unitary Cayley graph Xn = Cay(Z„, Un) with U„ = {x G Zn : gcd(x, n) = 1}. Here gcd(x, n) denotes the greatest common divisor of x and n and Un is the set of multiplicative units of Zn considered as a ring. In [15] we proved for n > 1 that x(Xn) = p, where p is a smallest prime divisor of n. Basic and Ilic [2] remarked in passing that Xn is uniquely p-colorable. This remark encouraged us to look closer at uniquely colorable Cayley graphs in general. In this paper we show that the chromatic number x(G) = k of a uniquely colorable Cayley graph G over a group r is a divisor of the number of elements |r| = n of r. Each color class of G is a coset of some subgroup of order n/k of r. For a uniquely colorable, noncomplete Cayley graph over an abelian group the estimate (1.1) on its number of edges can be improved to |E| > (k - 1)n. For every divisor k of n, 1 < k < n, we construct a uniquely k-colorable circulant graph on n vertices with the minimal number of (k - 1)n edges. In the final section, extending a result of Greenwell and Lovasz [10], we present a general method for constructing uniquely colorable graphs by graph products, which can especially be applied to Cayley graphs. 2 Necessary conditions A graph G = (V, E) is transitive if for any two vertices x, y G V there is an automorphism t of G with t(x) = y. Transitive graphs are regular. We call G weakly transitive if we require the existence of an automorphism t of G with t (x) = y only for adjacent vertices x and y. W Klotz and T. Sander: Uniquely colorable Cayley graphs 157 Lemma 2.1. Let the graph G = (V, E) be weakly transitive und uniqely k-colorable. Then x(G) = k is a divisor of | V| = n and every color class of G has n/k elements. Proof. We may assume k > 1. Let Ci, C2 be an arbitrary pair of color classes of G. Since x(G) = k there exists a pair x, y of adjacent vertices x G C and y G C2. As G is weakly transitive we know that there is an automorphism t of G with t(x) = y. Every automorphism of a uniquely colorable graph G maps each color class of G to another color class of G. Therefore, x G C1, y G C2 and t(x) = y imply t(C1) = C2 and |C1| = |C2|. Every color class C of G has the same number of |C| elements. As the color classes partition the vertex set V into k disjoint sets of equal size |C|, we have |V| = n = k|C|, which proves the lemma. □ Let G = Cay(r, S) be a Cayley graph. Define the bijection Ta : r ^ r for a G r by Ta (x) = xa. We verify for x, y G r: x, y adjacent in G ^ xy-1 G S ^ (xa)(ya)-1 G S ^ Ta(x),Ta(y) adjacent in G. For a = x-1y we have Ta(x) = y. This shows that H(r) = {Ta : a G r} is a subgroup of the automorphism group Aut(G) that operates transitively on the vertices of G. As Cayley graphs are transitive, Lemma 2.1 can especially be applied to Cayley graphs. Theorem 2.2. For a uniquely colorable Cayley graph G = Cay(r, S) the following statements are true. 1. The chromatic number x(G) = k divides the number |V(G)| = |r | = n of vertices of G. 2. Every color class C of G is a left coset of a subgroup U (C) C r of order |U (C) | = n k' 3. For any two distinct color classes C1 and C2 of G there exists an element y G r such that U(C2) = 7U(C1)y-1. If r is abelian, then every color class C of G has the same subgroup U(C). Proof. 1. This is a consequence of Lemma 2.1. 2. Suppose that C = {a1,..., ar }, r = n/k, is a color class of G. Define U = U(C) = {a-1aj : i,j G{1,...,r}}. We prove that U is a subgroup of r. The unit element e = a-1aj belongs to U. For x = a-1aj G U we have x-1 = a—1ai G U. Assume that x = a-1aj G U and y = a-1at G U. We are going to show xy G U. The automorphism tx of G maps a4 to aj, Tx(aj) = a4x = aj. From the unique colorability of G we conclude tx(C) = C and analogously Ty(C) = C. For arbitrary Z G C we have Tx(Z) = Zx = Z1 G C, Ty (Z1) = Z1y = Zxy = Z2 G C, xy = Z-1Z2 G U. Next, we show C = a1U, the left coset of U represented by a1. For every a4 G C we have aj = a1 (a-1ai) G a1U, which implies C C a1U. Suppose z G a1U, z = a1a—1aj- = a1x, x = a— 1aj- for some i, j G {1,..., r}. 158 Ars Math. Contemp. 12 (2017) 111-126 As above we see tx(C) = C. Therefore, z = a1x = rx(ai) G C, C = a\U. 3. Let C1 = aU1 and C2 = bU2 be different color classes of G, U1 U2 = U(C2). For the automorphism Td of G with d = a-1b we have Td(a) unique colorability of G implies Td(C1) = C2, hence C2 = C1d, bU2 = aU1 a-1b and therefore U2 = CU1C-1 with Z = b-1a. If r is abelian, we conclude U2 = U1. □ Corollary 2.3. If G = Cay(Zn, S) is a uniquely colorable circulant graph, then x(G) = k is a divisor of n. The color classes of G are the residue classes modulo k in Zn. If S is extended by elements s' G Zn, s' ^ 0 modulo k, to a self-inverse set S', then G' = Cay(Zn, S') is also a uniquely colorable graph with x(G') = k. Proof. According to Theorem 2.2, the color classes of G are the cosets of a subgroup U C Zn, |U| = n/k. The (additive) cyclic group Zn has exactly one subgroup of order n/k that is (k) = {0, k,..., (n/k — 1)k}, the cyclic subgroup generated by k. The cosets of (k) are the residue classes modulo k in Zn. The graph G' = Cay(Zn, S') is constructed from G by adding edges between different color classes. So the graph remains uniquely colorable with the same chromatic number. □ Problem 2.4. Is there a uniquely colorable Cayley graph over a nonabelian group such that different color classes are left cosets of different subgroups? Theorem 2.5. Let G = Cay(r, S) be a uniquely colorable Cayley graph over the abelian group r, |r| = n, x(G) = k < n. Then we have: The subgraph of G induced by any two color classes of G is uniquely colorable and regular of degree l > 2. Moreover, |E (G)| > (k — 1)n. This bound is sharp. Proof. The subgraph induced by any color classes of G must be uniquely colorable because otherwise G would not have this property. Consider arbitrary different color classes C and D of G. According to Theorem 2.2(3) they are cosets C = aU, D = bU of the same subgroup U = {u1,..., ur } C r, r = n/k. Without loss of generality let au1 be a vertex of maximum degree l in the subgraph G1 = G(C U D) induced by C U D in G. The neighbors of au1 in G1 must lie in bU. Let these be bu4l,..., bujj. For u G U we apply the automorphism tu of G defined by Tu(x) = xu to au1 and its neighbors in G1 and conclude: au1u G aU is adjacent to buil u,..., bu^ u G bU for every u G U. As au1u runs through all elements of aU for u G U, we see that all vertices in aU must have the same degree l in G1. The same holds for the vertices of bU since the r vertices of bU have rl edges in G1 and the maximum degree of G1 is l. It is easy to see (cf. Theorem 1 in [11]) that the subgraph G1 = G(C U D) induced by any two color classes C, D of G must be connected. This implies = U (C1), = b. The nn lT = |E (G1)|>|V (G1 )| — 1 = 2 - — 1 W Klotz and T. Sander: Uniquely colorable Cayley graphs 159 so that l > 2 - k > 1. n As l is an integer we have l > 2. This implies for |S|, the degree of regularity of G, |S| > 2(k - 1). Finally, we estimate the number of edges of G: |E(G)| = 2|S|n > (k - 1)n. Examples in the next section (see Corollary 3.4) will show that this bound is sharp. □ 3 Uniquely colorable Cayley graphs with few edges For the next theorem recall that the clique number "(G) of a graph G is the largest number of vertices in a complete subgraph of G. The clique number "(G) of the complementary graph G of G is also known as the independence number or stability number of G. Theorem 3.1. Let U be a subgroup of the (additive) abelian group r, |U | = | r | /k, k > 1 a divisor of |r|. Moreover, let jri,..., rk} be a system of distinct representatives of the cosets of U in r. Define S = {r - rj : i, j e {1,..., k}, i = j} and G = Cay(r, S). Then we have: 1. x(G) = "(G) = k. 2. x(G) = "(G) = JT. 3. The cosets of U in r are the color classes of an optimal coloring of G. Proof. From the definition of the representatives r1,..., rk we deduce Sn U = 0. Suppose that x, y belong to the same coset rj + U, 1 < i < k. Then we can find elements ui, u2 e U such that x = rj + u1 and y = rj + u2. Now x - y = u1 - u2 e U implies x - y e S, which means that x and y are not adjacent in G. The cosets of U partition the vertex set r of G into k stable sets, i.e. sets of pairwise nonadjacent vertices. So we have "(G) < x(G) < k. On the other hand r1,..., rk induce a clique of size k in G. This proves claims 1 and 3. Let U = {u1,..., ut}, t = |r|/k. The sets Kj = {rj + uj : i = 1,..., k}, 1 < j < t, induce cliques of size k in G, and therefore stable sets of size k in G. To show that these sets are pairwise disjoint, we assume x e Kj1 nKj2 for j1 = j2. We can find i1,i2 e {1,..., k} such that x ril ++ ujl ri2 ++ uj2 . Hence, rjl rj2 uj2 uji e U. 160 Ars Math. Contemp. 12 (2017) 111-126 From S n U = 0 we deduce i\ = i2, which implies j = j2 contrary to our assumption. The sets Kj, 1 < j < t, constitute a partition of the vertex set r of G into t = |r|/k stable sets of G. Therefore, we have "(G) < x(G) . Finally, claim 2 follows from the fact that every coset of U induces a clique of size t = |r|/k in G. □ Theorem 3.1 gives a first impression of what symbol sets may potentially yield uniquely colorable Cayley graphs. The next example, however, shows that the symbol set structure mentioned there is not sufficient in general for unique colorability. Example 3.2. We consider the integers modulo 12, r = Z12 = {0,1,..., 11}. Let U = (4) = {0,4, 8} be the cyclic subgroup of Z12 generated by 4. Then we have k = |r|/|U| = 4 and {r1, r2, r3, r4} = {0,1,6,7} as a system of distinct representatives for the cosets of U. We define S = {r - rj : i,j € {1,..., 4}, i = j} = {1, 5, 6, 7,11} and G = Cay(r,S). According to Theorem 3.1 the cosets of U in r, {0, 4, 8}, {1, 5, 9}, {2, 6,10}, {3, 7,11}, are the color classes of an optimal coloring of G. But there is another partition of Z12 into four stable sets of G: {0, 2,4}, {1, 3, 5}, {6, 8,10}, {7, 9,11}. Therefore, G is not uniquely colorable. A more careful choice of the system of representatives will improve the situation. Theorem 3.3. Let k be a divisor of n, 1 < k < n, Sk,n = {1, 2,..., k — 1} U {n — 1, n — 2,..., n — (k — 1)}, and Gk,n = Cay(Zn, Sk,n). Then the circulant graph Gk n is uniquely colorable with x(Gfc,„) = w(Gfc,n) = k and x(Gfc,„) = w(Gfc,n) = p (3.1) The residue classes modulo k in Zn are the maximal stable sets of Gk,n and the color classes of an optimal coloring of Gk,n. Proof. The integers ri =0, r2 = 1,..., rk = k - 1 constitute a system of distinct representatives for the cosets of the subgroup U = (k) generated by k in Zn. Modulo n we have: Sfc,n = {ri - rj : i, j G {1, 2, .. . ,k}, i = j}. Now Theorem 3.1 implies (3.1) and the fact that the cosets of U, i.e. the residue classes modulo k in Zn, are the color classes of an optimal coloring of Gk n. Let M be a stable W Klotz and T. Sander: Uniquely colorable Cayley graphs 161 set with a maximal number of vertices in Gk,n. We have |M| = n/k by (3.1). For every x G M the consecutive integers x +1,..., x + k - 1 (modulo n) are adjacent to x and therefore not in M. This implies that M is the residue class x + (k) in Zn. Let F be an optimal coloring of G„jfc, i.e. a coloring of the vertices of Gfcj„ with k colors. Every color class of F must be a maximal stable set of G„jfc with n/k elements. We have just shown that these sets are the cosets of U = (k) in Zn. Therefore, Gfcj„ is uniquely colorable. □ The graph Gfcj„ = Cay(Zn, Sk,n) is regular of degree |Sk,n| = 2(k - 1). This implies |E(Gfcj„)| = (k - 1)n. Hence we immediately obtain: Corollary 3.4. For every divisor k of n, 1 < k < n, the graph Gk,n defined in Theorem 3.3 is a uniquely k-colorable, circulant graph with n vertices and the minimal number of |E(Gfcj„)| = (k - 1)n edges. Example 3.5. Let Xn = Cay(Zn, Un) be the unitary Cayley graph on n vertices, Un = {x G Zn : gcd(x, n) = 1}. Suppose that p is the smallest prime divisor of n, 1 < p < n. According to Theorem 3.3 we define SPtn = {1, 2,... ,p - 1} U {n - 1, n - 2,..., n - (p - 1)} and GPj„ = Cay(Z„, Sp,n). Then Gp,n is uniquely colorable and x(Gp,n) = x(Xn) = p. The unitary Cayley graph Xn results from Gp,n by adding additional edges between different color classes of Gp,n. So Xn and Gp,n are both uniquely colorable with the same color classes. Problem 3.6. Is necessarily x(G) = w(G) for every circulant uniquely colorable Cayley graph? 4 Constructing uniquely colorable graphs by graph products The direct product X x Y of graphs X and Y has as its vertex set the cartesian product V(X) x V(Y). Vertices (xi, yi), (x2, y2) of X x Y are adjacent if xi is adjacent to x2 in X and y1 is adjacent to y2 in Y .If X = Cay(r1, S1) and Y = Cay(r2, S2) are Cayley graphs, then X x Y is a Cayley graph Cay(r, S) over the direct product r = r1 x r2 with shift set S = S1 x S2. A product X x Y of connected graphs is connected if both factors have at least two vertices and at least one factor is not bipartite (see [14]). Every proper n-coloring f : V (X) ^ Zn of X induces a proper n-coloring F : V (X) x V(Y) ^ Zn of X x Y by F(x, y) = f (x) for every x G V(X), y G V(y). As the same is true for Y instead of X, we immediately see x(X x Y) < min{x(X),x(Y)}. A famous conjecture of Hedetniemi ([12], [17]) states that always equality occurs. We denote by 2K2 the graph consisting of two disjoint edges. A graph X is 2K2-free if it has no induced subgraph 2K2. D. Turzik [18] showed that Hedetniemi's conjecture is true if one of the factors is 2K2-free. Lemma 4.1. Let the graph X be 2K2-free and let c : V(X) x V(Y) ^ Zn be a proper n-coloring of X x Y. For y G V (Y) define the map cy : V(X) ^ Zn by cy (x) = c(x, y) for every x G V(X). If every cy, y G V (Y), is an improper coloring of X, then x(Y) < n. 162 Ars Math. Contemp. 12 (2017) 111-126 Proof. The map cy is an improper coloring of X means that there are adjacent vertices xi, x2 of X such that cy(xi) = cy(x2). Let be the least value cy(xi) such that there are adjacent vertices x1, x2 of X with cy (x1) = cy(x2). We show that ^ is a proper n-coloring of Y. Let y1, y2 be adjacent vertices of Y. Assume ^(y1) = y>(y2). Then we find two pairs x1, x2 and x3, x4 of adjacent vertices in X such that cyi (x1) = cyi (x2 ) = ^(y1) = ^(y2) = cy2 (x3) = cy2 (x4), c(x1,y1) = c(x2,y1) = c(x3,y2) = c(x4,y2). (4.1) As x1 , . . . , x4 do not induce a subgraph 2K2 in X, either {x1, x2} n {x3, x4} = D = 0 or D = 0 and there is an edge between {x1, x2} and {x3, x4}. Suppose e.g. D = 0 and x1, x3 are adjacent. Then (x1, y1) and (x3, y2) are adjacent vertices of X x Y. But now c(x1, y1) = c(x3, y2) in (4.1) contradicts the fact that c is a proper coloring of X x Y. Similarly, the other cases lead to a contradiction. □ The following theorem extends a result of Greenwell and Lovasz [10]. Theorem 4.2. Let the graph X be uniquely n-colorable and 2K2-free. If Y is a connected graph with chromatic number x(Y) > n, then X x Y is uniquely n-colorable. Proof. We know x(X x Y) = m < x(X) = n. Let c : V(X) x V(Y) ^ Zm be an arbitrary proper m-coloring of X x Y. For y G Y define cy : V (X) ^ Zm by cy (x) = c(x, y) for every x G V(X). If cy is an improper m-coloring of X for every y G Y, then Lemma 2.1 implies x(Y) < m < n contradicting x(Y) > n. We conclude that there is a vertex y of Y such that cy is a proper m-coloring of X. Moreover, m < n = x(X) implies m = n. Let u be any neighbor of y in Y. Assume that there is a vertex x1 in X such that cu(x1) = cy(x1). As cy is a proper n-coloring of the uniquely n-colorable graph X, all n colors except cy (x1) appear in the range of cy at the neighbors of x1 . In particular, we find a neighbor x2 of x1 with cy(x2) = cu(x1), c(x2, y) = c(x1,u). But this is impossible, because (x2,y) is adjacent to (x1, u) in X x Y and c is a proper coloring of this graph. Therefore, we have cu(x) = cy(x) for every x G V(X). We may repeat the above argument for every neighbor of u. Continuing this way we reach every vertex in the connected graph Y and achieve the following result: c(x, y1) = c(x, y2) for every y1, y2 G V(Y) and every x G V(X). This implies that the color classes C1,..., Cn of the arbitrary n-coloring c of X x Y are given by the uniquely determined color classes D1,..., Dn of X, C = Dj x Y, for i = 1,..., n. This means that X x Y is uniquely n-colorable. □ In the following subsections we present some graph candidates for the application of Theorem 4.2. W Klotz and T. Sander: Uniquely colorable Cayley graphs 163 4.1 Complete multipartite graphs We call a graph X a complete m-partite graph if its vertex set V(X) can be partitioned into m nonempty, disjoint subsets ('color classes') such that each vertex is adjacent to every vertex which is not in his own class. Obviously, these graphs are uniquely m-colorable and 2K2-free. If a complete m-partite graph is regular, then all color classes must have the same size k. Such a graph can be represented as a Cayley graph over Zm x Zk. Corollary 4.3. Let X, be a complete m,-partite graph for i = 1,... ,r, r > 2, and 2 < mi < m2... < mr. Then X = Xi x X2 x ... x Xr has chromatic number x(X) = m1. The graph X is uniquely m1-colorable if and only if m1 < m2. Proof. We have x(X) < min{m1,..., mr} = m1. If we take one vertex from each color class of Xj we get a clique Q, of size m, in Xj. Assume that Q, has vertex set {1, 2,..., m,}. Then the tuples (a, a,..., a) with the r-fold entry a G {1, 2,..., m1} define a clique of size m1 in X. Thus we see x(X) = m1. If m1 < m2 we set Y = X2 x ... x Xr. This graph is connected with x(Y) = m2 > m1 = x(X1). Therefore, we may apply Theorem 4.2 to the product X1 x Y and conclude that it is uniquely m1 -colorable. If m1 = m2 = m, let f1 be an m-coloring of X1 and f2 be an m-coloring of X2. The colorings of X induced by f1 and by f2 are distinct optimal colorings of X. □ 4.2 Complementary graphs of compass graphs 0 14 13 12 9 Figure 1 4 5 The compass graph CS(k, P) is regular of degree 3 and has n = 6k vertices, k > 2. The vertices 0,1,..., n - 1 are arranged in this order along a hamiltonian cycle. Every vertex x divisible by 3 forms a triangle with the adjacent vertices x ± 1 mod n. By P we denote a partition of Zm = {0,1,..., m - 1}, m = 2k, in 2-element subsets which do not 164 Ars Math. Contemp. 12 (2017) 111-126 consist of two consecutive integers modulo m. For every {a, b} G P we connect the vertices 3a and 3b by an edge. Figure 1 displays CS(3, P) with P = {{0,3}, {1,4}, {2,5}}. Obviously, every compass graph CS(k,P) does not contain an induced cycle C4 of length 4. This means for the complementary graph CS(k, P) that it does not contain an induced 2K2. The maximal cliques of CS(k, P) are given by its triangles, which in CS(k, P) define the maximal stable sets. To achieve an optimal coloring of CS(k, P) we must take the sets of vertices {x, x - 1, x +1 mod n}, x = 0 mod 3, as color classes. The graph CS(k, P) is uniquely 2k-colorable. These graphs are candidates for the graph X in Theorem 4.2. It seems to be difficult to decide generally which compass graphs are Cayley graphs. The graph in Figure 1 is the only Cayley compass graph with 18 vertices. Similarly, we found that there is a unique Cayley compass graph with 12, 24, 42, 48 or 54 vertices. But there is definitely no such graph with 30 or 36 vertices. Again, we found a compass graph with 60 vertices, which is a Cayley graph over the alternating group A5. But we do not know if it is unique. Infinite sequences of 2K2-free, uniquely colorable Cayley graphs can be constructed by the following operations. The k-fold join, join(k, G), of a graph G consists of k disjoint copies Gi,..., Gk of G. For every i < j every vertex of Gj is connected by an edge to every vertex of Gj. Let the n x n-matrix A be an adjacency matrix of G and Jk the k x k-matrix with all entries equal to 1. The Kronecker product Jk x A is the (kn) x (kn)-matrix which results from Jk by replacing every entry by A. The k-fold clone, clone(k, G), is the graph with adjacency matrix Jk x A. We leave the proof of the following statement as an exercise for the reader. Proposition 4.4. If the Cayley graph G is 2K2-free and uniquely colorable then join(k, G) and clone(k, G) are 2K2-free, uniquely colorable Cayley graphs for every integer k > 2. References [1] S. Akbari, V. S. Mirrokni and B. S. Sadjad, Kr-free uniquely colorable graphs with minimum possible edges, J. Comb. Theory, Ser. B 82 (2001), 316-318. [2] M. Basic and A. Ilic, On the chromatic number of integral circulant graphs, Comput. Math. Appl. 60 (2010), 144-150. [3] D. Bayer, The division algorithm and the Hilbert scheme, PhD thesis, Harvard University, 1982. [4] B. Bollobas, Uniquely colorable graphs, J. Comb. Theory, Ser. B 25 (1978), 54-61. [5] C. Y. Chao and Z. Chen, On uniquely 3-colorable graphs, Discrete Math. 112 (1993), 21-27. [6] G. Chartrand and D. P. Geller, On uniquely colorable planar graphs, J. Comb. Theory 6 (1969), 271-278. [7] A. Daneshgar, Forcing structures and cliques in uniquely vertex colorable graphs, SIAM J. Discrete Math 14 (2001), 433-445. [8] A. Daneshgar and R. Naserasr, On small uniquely vertex colorable graphs and Xu's conjecture, Discrete Math. 223 (2000), 93-108. [9] C. Godsil and G. Royle, Algebraic graph theory, Graduate Texts in Mathematics, Vol 207, Springer, 2001. [10] D. Greenwell and L. Lovasz, Applications of product coloring, Math. Acad. Sci. Hungar. 25 (1974), 335-340. W Klotz and T. Sander: Uniquely colorable Cayley graphs 165 [11] F. Harary, S. T. Hedetniemi and R. W. Robinson, Uniquely colorable graphs, J. Comb. Theory 6 (1969), 264-270. [12] S. Hedetniemi, Homomorphisms of graphs and automata, Technical report university of Michigan 03105-44-T (1966). [13] C. J. Hillar and T. Windfeldt, Algebraic characterization of uniquely vertex colorable graphs, J. Comb. Theory, Ser. B 98 (2008), 400-414. [14] R. Hammack, W. Imrich and S. KlavZar, Handbook of product graphs, 2nd ed., CRC Press, 2011. [15] W. Klotz and T. Sander, Some properties of unitary Cayley graphs, Electron. J. Comb. 14 (2007), R45, 1-12. [16] L. Lovasz, Stable sets and polynomials, Discrete Math. 124 (1994), 137-153. [17] C. Tardif, Hedetniemi's conjecture 40 years later, Graph Th. Notes New YorkLIV(2008), 46-57. [18] D. Turzik, A note on direct product of graphs, Comment. Math. Univ. Carolin. 24 (1983), 461463. [19] S. Xu, The size of uniquely colorable graphs, J. Comb. Theory, Ser. B 50 (1990), 319-320. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 167-182 Non-negative spectrum of a digraph Irena M. Jovanovic * School of Computing, Union University, Knez Mihailova 6, 11000 Belgrade, Serbia Received 8 June 2014, accepted 29 March 2016, published online 26 November 2016 Digraphs are considered by means of eigenvalues of the matrix AAT, and similarly ATA, where A is the adjacency matrix of a digraph. The common spectrum of these matrices is called non-negative spectrum or N-spectrum of a digraph. Several properties of the N-spectrum are proved. The notion of cospectrality is generalized, and some examples of cospectral (multi)(di)graphs are constructed. Keywords: Digraph, non-negative spectrum, multigraph, cospectrality, isomorphism. Math. Subj. Class.: 05C20 1 Introduction Spectral (di)graph theory means usage of linear algebra tools and techniques in the study of (di)graphs. It is a very well developed mathematical field (see [8] or [6]) with many applications (see, for example, [2] and [15]). For any (di)graph matrix M, one can build a spectral (di)graph subtheory, and then be able to study (di)graphs by means of eigenvalues of the matrix M. We will denote these eigenvalues M-eigenvalues. In general case, in order to avoid confusion, to any notion in the corresponding subtheory a prefix 'M' should be added. Frequently used graph matrices are the adjacency matrix A, the Laplacian L = D - A and the signless Laplacian Q = D + A, where D is a diagonal matrix of vertex degrees. The spectral (di)graph theory then consolidates all these particular subtheories together with interaction tools. In this paper, digraphs are considered by means of eigenvalues of the matrix AAT, and similarly ATA, where A is the adjacency matrix of a digraph. The common spectrum of these matrices is denoted N-spectrum and called non-negative spectrum of a digraph. According to [5], the N-spectrum of a digraph was not considered in the mathematical *The work is supported by Serbian Ministry of Education, Science and Technological Development, Projects 174033 and 11145003. E-mail address: irenaire@gmail.com (Irena M. Jovanovic) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 168 Ars Math. Contemp. 12 (2017) 111-126 literature so far. Since the matrices AAT and ATA appear in applications (see, for example, [11] and [12]), we believe that introduced notion and presented results could be useful to mathematicians and informaticians. Namely, N-spectrum can facilitate the examination of digraphs since frequently used adjacency matrix of a digraph is not symmetric in general case, and therefore its spectrum consists of complex numbers. It is well known that digraphs serve as models for different processes and phenomena in computer sciences, where some spectrally based techniques are used in investigations. By this approach some new conclusions and comparisons of existing results could be made. The paper is organized as follows: In Section 2 basic digraph terminology is given and some elementary facts related to the matrices AAT and AT A and their spectrum are pointed out. Since this paper represents the first mathematical paper on the N-spectrum, elementary observations useful for further work are presented in Section 3. In Section 4 the effect of certain digraph operations and transformations on the N-spectrum is studied. One family of N-cospectral digraphs is determined in this section. Structural similarity (i.e. values and layout of entries in the matrix) of the matrix AAT of some digraph with the adjacency or the signless Laplacian matrix of some multigraph, has motivated us to generalize the notion of cospectrality in Section 5. The study of cospectrality with respect to different (multi)(di)graph matrices could be useful in finding connections between different spectral subtheories that are based on these matrices, and, what is more important, in finding new pairs of cospectral (multi)(di)graphs in particular spectral subtheory. That way, certain pairs of multigraps that are cospectral with respect to the adjacency matrix are found. The study of spectral subtheory based on the signless Laplacian matrix is currently used (see, for example, [7]), so the paper is concluded with some examples of digraphs and multigraphs whose N- and Q-spectrum, respectively, are the same. 2 Preliminaries Let D = (V (D), E (D)) be a digraph of order n with the set of vertices V (D) = {v1, v2, ..., vn}. The set of edges E(D) consists of ordered pairs of vertices, and we suppose that the loops, i.e. the edges of the form (vj, vj) are permitted, but multiple edges are not. The adjacency matrix A = [ajj ] of D is the binary matrix of order n, such that ajj = 1, if there is an edge from vj to vj, and otherwise ajj = 0. If e = (vj, vj) is the edge of D, we say that vj is the initial vertex of e, while vj is the terminal vertex. The vertex vj G V(D) is the out-neighbour of the vertex vj G V(D) if there is the edge (vj, vj) g E(D). The vertex vk G V(D) is the in-neighbour of the vertex vj G V(D) if there is the edge (vk, vj) G E(D). The out-degree of vertex vj, denoted by outdegD(vj) or dD(vj), is the number of edges of which it is the initial vertex, while the in-degree of vj, denoted by indegD (vj) or dD (vj), is the number of edges of which vj is the terminal vertex. A loop at some vertex contributes 1 to both the in-degree and the out-degree of that vertex. Let us suppose that the edges of D are ordered as ei, e2,..., em. The in-incidence matrix of D is the n by m matrix Bjn = [bjj] such that bjj = 1 if ej = (vk, vj) for some vertex vk, and otherwise bjj = 0. The out-incidence matrix Bout = [gjj] of the digraph D is the n by m matrix such that gjj = 1 if ej = (vj, vi) for some vertex v;, and otherwise gjj = 0. It is a matter of routine to check that A = BoutBjn holds. The characteristic polynomial det(XI - A) of A is the characteristic polynomial of the I. M. Jovanovic: Non-negative spectrum of a digraph 169 digraph D, and the eigenvalues of A are the eigenvalues of D. For the remaining notation and terminology related to digraphs, and also graphs, we refer the reader to [5], [2], [3], [1], [8] and [6]. In this paper we are interested in the structural characteristics of a digraph D related to the spectrum of matrices AAT and AT A, where A is the adjacency matrix of D. The matrices AAT and ATA are non-negative, square and symmetric. One can easily check that these matrices are positive semi-definite (see, for example, [14]), which means that their eigenvalues are non-negative. The entries of the matrices AAT and AT A are characterised by the following proposition (see [12]): Proposition 2.1. The (i,j)-entry of the matrix AAT (AT A) of D is equal to the number of common out-neighbours (in-neighbours) of vi and Vj. Diagonal entries of the matrix AAT (AT A) represent out-degrees (in-degrees) of the vertices of D. According to the previous observations, one can introduce the following notation: Nout = AAT and Nin = ATA. The characteristic polynomial det(AI — Nin) of Nin is the Nin-characteristic polynomial of D, while the characteristic polynomial det(AI - Nout) of Nout is the Nout-characteristic polynomial of D. Since the spectrum of Nout and Nin is the same (see [14]), it can be denoted by the single name - the N-spectrum. Therefore, the characteristic polynomials N(x) of these matrices can be named the N-polynomials. However, we underline that through the investigation we mainly considered Nout(D) matrix of D, whose spectrum is denoted by ni > V2 > ... > Vn. The N-spectral radius pN(D) of D is defined to be the spectral radius of Nout(D), and similarly Nin(D). Remark 2.2. For the N-spectrum n1,V 2,... ,Vn of a digraph D with m edges the following holds: • The numbers n1, V 2,... ,Vn are real and non-negative, • Vi + V2 + ... + Vn = m, • D is consisted of only isolated vertices if and only if n1 = n2 = ... = Vn =0. 3 Some basic results In this section we give some elementary results that we will use in the subsequent sections. Let us remind you that a digraph D is r-regular if the in-degree and the out-degree of each its vertex are equal to r. By use of the basic combinatorial principles for counting one can easily check that the row sum for each row of the matrix Nout(D) is equal to r + r(r — 1) = r2. Now, we can prove the following lemma: Lemma 3.1. N-spectral radius pN (D) of a r-regular digraph D of order n is r2. Proof. Since Nout(D) is the square, non-negative matrix with equal row sums, according to Theorem of Frobenius (see [4]) the spectral radius of this matrix is r2. □ Remark 3.2. The eigenvector that corresponds to the N-eigenvalue r2 of a r-regular digraph D is all-1 vector. 170 Ars Math. Contemp. 12 (2017) 111-126 o Example 3.3. The complete digraph of order n is the digraph Kn in which for each pair of vertices there is an edge, including a loop at each vertex. The N-characteristic polynomial of this digraph is: Ng (x) = (x - n2)xn-1, and thus its N-spectrum is: n2, [0]n-1. Here, and in the further text, an eigenvalue n of the multiplicity k is denoted by [n]k. Let us now consider connected digraphs whose vertices do not have the common out-neighbours. If D = (V(D), E(D)) is such a digraph, then indegD(vi) < 1 must hold for each vertex vi e V(D). Let us remind you that a rooted oriented tree, briefly rooted tree, is an oriented tree with a specific vertex v1, called the root, such that for every other vertex vj the path connecting v1 to vj is a directed path from v1 to vj. This means that D is connected, indeg(v1) = 0 and indegD (vi) = 1 for every other vertex vi of D, and vice versa according to Theorem 15.2 from [1]. It is obvious that vertices of a rooted tree do not have the common out-neighbours. If in a digraph D whose vertices do not have the common out-neighbours there are at least two vertices such that their in-degrees are equal to 0, D would not have been connected, i.e. D would consist of at least two connected components. Since in a rooted tree there is unique vertex v1 such that indeg(v1) = 0, one can add one extra edge to obtain a digraph where there is no pair of vertices with common out-neighbours. We distinguish two possibilities: this extra edge is a loop at v1, i.e. (v^^/U])) or it is an edge (vx, v1), for exactly one vertex vx of a rooted tree. Hence, we can say that a resulting digraph is a unicyclic digraph derived from a rooted tree (Figure 1). Figure 1: Unicyclic digraphs whose vertices do not have the common out-neighbours That way, the following proposition is proved: Proposition 3.4. D is a connected digraph whose vertices do not have the common out-neighbours if and only if it is a rooted tree or a unicyclic digraph that can be derived from a rooted tree. Remark 3.5. Since the matrix Nout(D) of a connected digraph D such that there is no pair of vertices with the common out-neighbours in D is the diagonal matrix of vertex degrees, the N-spectrum of D is: outdegD(vi), outdegD(v2),..., outdegD(vn). I. M. Jovanovic: Non-negative spectrum of a digraph 171 Remark 3.6. The converse digraph Conv(D) of a digraph D is obtained by reversing the direction of each edge of D (see [2]). So, a digraph whose vertices do not have the common in-neighbours is the converse digraph of a rooted tree or of a unicyclic digraph that can be derived from a rooted tree. Example 3.7. The N-characteristic polynomial of a rooted tree D is: Nd(x) = xl J^J (x — outdeg(vi)), ViEU (D) where l is the number of vertices vx such that outdegD(vx) = 0, while U(D) c V(D) is the set of vertices whose out-degree is at least 1. The digraph Pn is the special case of a rooted oriented tree. If V(Pn) = {vi, v2,..., vn } is the set of vertices of this digraph, then its set of edges consists of the pairs of vertices (vi, vi+1), for i = 1, 2,...,n — 1. The N-characteristic polynomial of Pn is: N^ (x) = x(x — 1)n-i. 1-regular digraph Cn is the special case of a unicyclic digraph derived from a rooted tree. Its N-characteristic polynomial is: N^ (x) = (x — 1)n. (3.1) 4 Some digraph operations and transformations We open this section with the result related to the N-spectrum of the complement of a given regular digraph. The complement DC = (V(DC),E(DC)) of a digraph D = (V(D),E(D)) has the vertex set V(DC) = V(D) and e e E(DC) if and only if e e E(D). Also, there is a loop at vertex vi in DC if and only if there is no loop at vi in D. Similarly to the proof of Theorem 2.1.2 from [6] for regular graphs we can prove the following: Proposition 4.1. If the N-eigenvalues of a r-regular digraph D of order n are ni(D), i = 1,2,...,n, then the N-eigenvalues of DC are n1(DC) = (n — r)2 and ni(DC) = ni(D), i = 2, 3, ... ,n. Proof. If Ad is the adjacency matrix of D and J is all-1 matrix, we find: Nom(DC ) = J2 — Ad J — J AD + Ad AD = (n — 2r)J + Nout(D), because the row sum for each row of AD is equal to r. □ Let us denote by D the digraph obtained from a connected digraph D by deleting the edge (vi, vj). Then we have: Nout(D) = Nout(D ) + M. Here, M = [mpq] is the square matrix of order n such that mii = 1 and mii = mii = 1 for each pair of vertices vi, vi such that (vi ,vj), (vi ,vj) e E (D), where l e {1, 2,..., n} \ {i}. Theorem 4.2. (Interlacing theorem - edge version) Let D be a connected digraph of order n whose N-spectrum is n1(D) > n2(D) > • • • > nn(D), and there is at least one 172 Ars Math. Contemp. 12 (2017) 111-126 vertex vj in D such that indeg D (vj) = 1. Let D be a digraph obtained from D by deleting an edge (vi, vj). If the N-eigenvalues of D are ni(D ) > (D ) > • • • > nn(D ), then m(D) >ni(D') > m(D) > m(iD) > ...Vn(D) > Vn(D') > 0. Proof. Since the spectrum of the matrix M consists of [1] and [0]n-1, the proof follows from Courant-Weyl inequalities (see, for example [6]). □ Remark 4.3. By considering Nin matrix of a digraph, one can prove that the previously given Interlacing theorem holds also for a connected digraph D in which there is at least one vertex vj such that outdegD (vj) = 1, and for its subdigraph D obtained from D by deleting an edge (vj,vi), for some vertex vi. In general case, such the N-eigenvalue interlacing does not hold. Namely, we have the following example. Example 4.4. For the digraph D that is depicted on Figure 2 and the digraph D that is obtained from D by deleting the edge (1,3), the N-interlacing property holds, i.e. for the N-spectra of these digraphs we have the following inequalities: 4.390 > 3.879 > 1.838 > 1.653 > 1 > 1 > 0.544 > 0.468 > 0.228 > 0. On the other hand, the N-eigenvalues of the digraphs D1 (Figure 2) and D1, that is obtained from Di by deleting the edge (1, 3), are 5.303 > 1.697 > 1 > 1 > 0, and similarly 4.115 > 1.764 > 1 > 1 > 0.139, so the N-interlacing property does not hold in this case. Figure 2: Digraphs D and D1 from Example 4.4 Now, we will consider a digraph D* obtained from a connected digraph D by adding a pendant edge at the vertex vi of D (i.e. an edge of the form (vx, vi) such that indegD* (vx) = 0 and outdegD* (vx) = 1, or an edge of the form (vi, vx) such that indegD»(vx) = 1 and outdego* (vx) = 0). The following statement obviously holds. Proposition 4.5. Let D* denotes a digraph obtained from a connected digraph D of order n by adding a pendant edge (vn+1, vi) at the vertex vi such that indegD (vi) = 0. Then the N-characteristicpolynomial of D* is: ND* (x) = (x — 1)No(x). □ Let us denote by Dvk a digraph obtained from a digraph D by deleting the vertex vk, and let (vi,vj-) = 1, if (vi,vj-) G E(D), and otherwise (vi,vj) = 0, for i,j e {1, 2,..., n}. I. M. Jovanovic: Non-negative spectrum of a digraph 173 Definition 4.6. The digraph D™^ v ) is the out-(vk, vj)-shrinking of D if for the edge (vfc, vj) in E(D), V(D^vo) = T^'vDD^vfc) and E^U)) = E(Dvfc) U {(vj,Vj)|MD(vj,vfc) = 1, for each j = k}. It is obvious that D°Ut v.) is a multidigraph in general case, and that if indegD (vj) = 1 then the matrix No„i(D?Vfci v,)) equals the matrix obtained from Noui(D) by deleting the k-th row and the k-th column. Theorem 4.7. Let D* denotes a digraph obtained from a connected digraph D of order n by adding the pendant edge (vj,vj) at the vertex vj such that (vk,vj) G E(D) and rndegD (vj) = 1. Then Nd, (x) = (x - 1)Nd(x) - N^o-t (x), where ND°«t (x) is the N-characteristic polynomial of the digraph D™^ v ) that is the out-(vk, vj)-shrinking of a digraph D. Proof. Since indegD* (vj) = 2, we have NOMi(D) r NOMi(D*)= 1 ' (n+1)x(n+1) where r = (0,..., 0, 0,..., 0)T is the vector of order n. The only no null coordinate k of the vector r corresponds to the common out-neighbour of vk and vj. By expanding the determinant of the matrix x/ - Noui(D*) by the last row we get: Nd (x) = det (x/ - Nout(D*)) = (x - 1)ND(x) + (_1)(n+1)+k • det (M|r), where the matrix M is obtained from x/ - Noui(D) by deleting the k-th column. Now, by expanding the determinant of the matrix (M|r) by the last column, we have: Nd* (x) =(x - 1)Nd(x) + (-1)(n+1)+k (-1)k+n det (x/ - M= (x - 1)Nd(x) - det ^x/ - M, where M is obtained from the matix Noui(D) by deleting the k-th row and k-th column. □ The line digraph L(D) of a digraph D (see, for example [5]) is the digraph whose vertices are the edges e1, e2,..., em of D such that there is an edge from ej to ej in L(D) if and only if the terminal vertex of ej equals the initial vertex of ej in D. If an edge ep is a loop at some vertex of D, then it becomes a loop at ep in L(D). Some results on adjacency spectra and energies of iterated line graphs are exposed in [13]. On the similar way, we can define iterated line digraphs. If D = L0(D) is a digraph and L(D) = L1 (D) is its line digraph, then Lk (D), k = 2, 3,... defined recursively by the formula Lk (D) = L (Lk-1(D)) are the iterated line digraphs of D. The line digraph of an r-regular digraph is also r-regular digraph. More precisely, the line digraph L1 (D) of an r-regular digraph D of order n is the r1 = r-regular digraph of order n1 = nr. Consequently, Lk (D), k = 2,3,... is the rk = r-regular digraph of order nk = rnk-1 = rkn, where nk-1 is the order of the digraph Lk-1 (D). 174 Ars Math. Contemp. 12 (2017) 111-126 Theorem 4.8. The N-eigenvalues of the line digraph L(D) of a r-regular digraph D are: [r2]n, [0](r-1)n. Proof. We will determine the N-characteristic polynomial NL(D) of L(D) related to the Nout(L(D)) matrix. As L = BfnBout is the adjacency matrix of the line digraph L(D) of D (see [5]), where Bin and Bout are the in-incidence matrix and the out-incidence matrix of D, respectively, we find: Nout(L(D)) = rBfnBin. Here, we have that the diagonal matrix whose entries are the out(in)-degrees of vertices in D is: A = rl = BinBT = BoutBTut. According to Lemma 8.2.3. from [10] we get: det (I - BinBin) = det (I - BTinBin), det (In - x-1rIn) = det (im - x-11Nout(L(D)^ . det (xI„ - rIn) = de^xIm - 1 Nout(L(D))^ , i.e. 1 Furthermore we have: and also det ((x - r + 1)/„ - In) = xn-m — det (rx/m - Nout(L(D))). According to (3.1) we find: N- (x - r + 1) = xn-mrmNL{D)(rx), i.e. Nl(d)(x) = xm-n(x - r2)n, and the proof follows. □ Therefore the N-spectrum of the line digraph Lk (D) of a r-regular digraph D of order n consists of [r2]nfc = [r2]nr and [0](r-1)nk = [0](r-1)r n, and hence we have the following corollary: Corollary 4.9. Let D1 and D2 be two r-regular digraphs of order n (not necessary N-cospectral). Then for all k > 1 digraphs Lk (D1) and Lk (D2) are N-cospectral. This way, we found a family of N-cospectral mates (i.e. the digraphs whose N-spectra are the same). We will continue examination of cospectrality in the next section. 5 Cospectrality relation Let DM be the set of (multi)(di)graphs D of order n with the associated spectrum aM (D) related to some (multi)(di)graph matrix M. Let us introduce the relation p C DM x DM2 between sets DM and DM2, for some (multi)(di)graph matrices M1 and M2 in the following way: we say that the (multi)(di)graph D1 is in the relation p with the (multi)(di)graph D2, i.e. D1pD2 if and only if aMl (D1) = aM2 (D2). So, the relation p is the cospectrality relation, while D1 and D2 form an (M1; M2) -cospectral mate. That way, we can generalize the notion of cospectrality: I. M. Jovanovic: Non-negative spectrum of a digraph 175 Definition 5.1. Let M1 and M2 be some (multi)(di)graph matrices. If the (multi)(di)graph D1 G DM is in the cospectrality relation p with the (multi)(di)graph D2 G D^2, i.e. the M1-spectrum of a (multi)(di)graph D1 is equal to the M2-spectrum of a (multi)(di)graph D2, then D1 and D2 are called (M1, M2)-cospectral (multi)(di)graphs. It is obvious that p is the equivalence relation on the set DM, in which case (multi)(di)-graphs D1 and D2 such that D1pD2 are M-cospectral. As a result of the composition of the cospectrality relations, one can get some new pairs of cospectral (multi)(di)graphs, as follows. Let us consider the set D^ of digraphs D of order n with the associated N-spectrum <7n (D). Clearly, N is related to Nout or Nin matrix of a digraph. Let us denote by GA+ and GA- the sets of out-multigraphs and in-multigraphs, respectively with the corresponding adjacency spectra. The in-multigraph M- g GnA- and the out-multigraph M+ G GA+ are associated to a digraph D g D^ in the following way: Definition 5.2. The in-multigraph M- = (V (MD), E(M-)) of adigraph D is the multigraph such that V(M-) = V(D), {vi,vj} G E(M-) if and only if there is a vertex vk G V(D) such that (vk, vi), (vk, vj) G E(D), and for each edge (vk, vi) in D there is a loop at vi in MD- . Definition 5.3. The out-multigraph M+ = (V (M+),E(M+)) of a digraph D is the multigraph such that V(M+) = V(D), {vi, vj} G E(M+) if and only if there is a vertex vk such that (vi, vk), (vj, vk) G E(D), and for each edge (vi, vk) in D there is a loop at vi in M+ . According to the previous definitions, one can notice the cospectrality relation, say p-, between sets GA- and DN, and similarly the cospectrality relation, say p+, between sets DrN and GA+. As the result of the composition of relations p+ and p- the pairs of A-cospectral multigraphs M- and M+ are getting. That way we have: Theorem 5.4. Multigraphs M- and M+ are A-cospectral. So, the exposed construction is a way for obtaining new pairs of cospectral and not necessarily isomorphic multigraphs. Example 5.5. The adjacency matrix of the in-multigraph M-, and similarly the outmultigraph M+, that is associated to the digraph D (which is depicted on Figure 3) is: A(Md) = Nin(D) : / 2 0 2 0 \ 0 110 2 13 0 ^ 0 0 0 1 ) , and A(M+) = Nout(D) = /2101 \ 1202 0 0 1 0 1202 Remark 5.6. Multigraphs M- and M+ associated to a digraph D are simple graphs only in the case when digraph D is a set of isolated vertices. If we permit existence of single loops (i.e. loops of multiplicity one) in a simple graph, the primary digraph D can be Cn or Pn. In this case, multigraphs M- and M+ are the sets of isolated loops or the disjoint unions of isolated loops and a single isolated vertex, and therefore M- and M+ are not only A-cospectral but also isomorphic. 176 Ars Math. Contemp. 12 (2017) 111-126 There are many examples where the multigraphs M— and M+ associated to a given digraph D are isomorphic, so the investigation of such multigraphs can be the subject of future research. If a primary digraph D is such that if (vj, vj) G E(D) then also (vj, vj) G E(D), for all vj, vj G V(D), it is obvious that the associated multigraphs M— and M+ will be isomorphic. We also have: Proposition 5.7. Multigraphs M— and M+ associated to a digraph D of prime order, n > 2, with circulant adjacency matrix are isomorphic. Proof. Since Njn(D) and Nout(D) are circulant matrices with the same eigenvalues, according to Theorem 1 from [9] they are permutationally similar. □ For an integer n > 2 and a set S C {1,2,..., n - 1} the circulant digraph Cn(S) is a digraph such that V(Cn(S)) = {1, 2,..., n} and E(C„(S)) = {(i, i + j (modn)) : 1 < i < n, j G S}. Circulant digraphs are of great interest in the graph and digraph theory and their applications (see [2]). Proposition 5.8. isomorphic. Multigraphs MD and M+ associated to a circulant digraph Cn(S) are Proof. Since the converse digraph Conv(Cn(S)) of Cn(S) is isomorphic to Cn(S) (according to Proposition 2.14.1 from [2]) and since Njn(Cn(S)) = Nout(Conv(Cn(S))), and similarly Nout(Cn(S)) = Njn(Conv(Cn(S))), the proof follows. □ Example 5.9. The matrix Nout (D) of the 2-regular digraph D that is depicted on Figure 4 structurally corresponds to the signless Laplacian matrix Q(M) of the 2-regular graph M, also depicted on Figure 4, i.e. /2 110 \ / 0 1 1 0 \ /2000 \ 1201 _ 1001 + 0200 1021 _ 1001 + 0020 \ 0 1 1 ^ \011^ \0002y Nout(D) _ _ Q(M). I. M. Jovanovic: Non-negative spectrum of a digraph 177 That way, one can notice the cospectrality relation p C DN x GQ between set DN of digraphs D of order n with the associated N-spectrum aN (D) and the set GQ of multigraphs M of order n with the associated Q-spectrum ctq (M). Figure 4: Triplet of (N, A, Q)-cospectral digraph D, multigraph MD = M+ and graph M, respectively This one and similar examples have motivated us to examine some new (N, Q)-cospectral mates. Furthermore, the multigraph M that makes (Q, N)-cospectral mate with a given digraph D can be used in determining some isomorphic multigraphs M- and M+, as follows: Proposition 5.10. Let D be a connected r-regular digraph of order n. If Nout(D) = Q(M ) holds for some multigraph M, then r = 0 or r = 2. Proof. We have Nout(D) = ri + C, where row sum of C is r(r - 1) for each row. If Nout(D) is the signless Laplacian matrix of some multigraph without loops, then r = r(r — 1) holds, which implies r = 0 or r = 2. On the other hand, if Nout(D) is the signless Laplacian matrix of a multigraph with loops, then the number of loops at some vertex is (r — (r — 1)r)/2, which means that r = 0 or r = 2. □ Remark 5.11. The statement from the previous proposition also holds in the case of the matrix Nin(D). Beside that, the multigraph M is the connected r-regular multigraph without loops. Therefore, we conclude that multigraphs M- and M+ associated to some 2-regular digraph D are isomorphic. In order to examine (N, Q)-cospectrality, we will introduce some binary digraph operations. Still, according to the nature and the mutual relationships between entries of matrices Nout(D) and Q(M) of some digraph D and some multigraph M, respectively, one can suspect poor variety in terms of the structure and the order (i.e. number of vertices) of the (N, Q)-cospectral mates (that could be obtained by direct comparing of these matrices). Let Di = (V(Di), E(Di)) and D2 = (V(D2), E(D2)) be two disjoint digraphs (i.e. digraphs with no common vertices nor edges). Definition 5.12. The out-join D1 VouiD2 of two disjoint digraphs D1 = (V (D1), E(D1 )) and D2 = (V(D2),E(D2)) is the digraph D = (V(D),E(D)) such that V(D) = 178 Ars Math. Contemp. 12 (2017) 111-126 V(Di) U V(D2) and E(D) = E(Di) U E(D2) U {(u,v)|u e V(Di),v e V(D2)}, for each u e V(D1) and v e V(D2). It is obvious that this digraph operation is not commutative, i.e. D1 VoutD2 = D2 Vout D1. Nout(D) matrix of the digraph D which is obtained by out-joining is: where A1 and A2 are the adjacency matrices of digraphs D1 and D2, respectively, while J is all-1 matrix. Each entry of the j-th row of the matrix A2 JT is equal to outdegD2 (uj), where uj e V(D2). In the same way one can define: Definition 5.13. The in-join D1VinD2 of two disjoint digraphs D1 = (V(D1), E(D1)) and D2 = (V(D2),E(D2)) is the digraph D = (V(D),E(D))such that V(D) = V(D1) U V(D2) and E(D) = E(D1) U E(D2) U {(v,u)|v e V(D2),u e V(D1)}, for each u e V(D1) and v e V(D2). Definition 5.14. The join D1VD2 of two disjoint digraphs D1 = (V(D1),E(D1)) and D2 = (V(D2), E(D2)) is the digraph D with the vertex set V(D) = V(D1) U V(D2), whose set of edges is E (D) = (E^V^t D2) U E ^^¿^2)) \ (E (D1) U E(D2)). Proposition 5.15. Let D = D1 VoutD2 be the digraph obtained by out-joining two connected disjoint digraphs D1 and D2 of orders n1 and n2, respectively. If Nout(D) = Q(M) holds for some multigraph M, then: 1. D1 is an isolated vertex, while D2 is a unicyclic digraph derived from a rooted tree. (a) if «4 = 1, then D =K^ (b) if n1 = 2, then D1 is any of digraphs depicted on Figure 5, (c) if n1 = 3, then D1 is 1-regular digraph, (d) if n1 > 4, then there is no digraph D1 such that the statement given by the proposition holds. Proof. Let us denote by V(D1) = {u1, u2,..., uni} and V(D2) = {v1, v2,..., vn2} the sets of vertices of digraphs D1 and D2, respectively. If Nout(D) = [njj] is the signless Laplacian matrix of some multigraph M, then by observing its rows n1 + 1, n1 + 2,..., n1 + n2, one can conclude that the number: 2. D1 =K1, while D2 is a rooted tree; 3. D2 is an isolated vertex, and: (1 - «1) outdegD2 (vp) - npq(NOMi(D2)), q=1,q=p I. M. Jovanovic: Non-negative spectrum of a digraph 179 Figure 5: Digraphs from Proposition 5.15 for each p = 1,2,..., n2, is zero or even positive integer. This means that n1 = 1 and D2 is a digraph such that there are no vertices with the common out-neighbours or D2 is an isolated vertex. In the former case, by observing rows 1, 2,... ,n1 of Nout(D), one concludes that: outdegDl (uk) + n2 - m2, for each k = 1, 2,... ,n1, is zero or even positive integer. Here m2 is the number of edges of D2, and the proof for statements 1. and 2. follows. If D2 is an isolated vertex, then by observing rows 1, 2,... ,n1 of Nout(D), we get that: outdegDl (uk) - ^ nkt(Nout(Di)) - nx +2, l=l,l = k (5.1) for each k = 1,2,..., ni, is zero or even positive integer. Let us consider the structure of Di. If n1 = 1 or n1 = 2, statements (a) and (b) follows from (5.1) by direct computation. If n1 = 3, then 3 > outdegDl (uk) > 1 must hold for each k = 1, 2,3. Let us suppose that outdegD1 (ui) = 3. This implies indegD1 (ui) = indegD1 (U2) = indegD1 (u^) = 1, and since the out-degree of u2 and u3 must be at least 1, (5.1) will be a negative number for at least one k. One can analyse the case when outdegDl (u1) = 2 the same way. And finally, if outdegDl (u1) = 1, (5.1) is a non-negative integer if and only if J23=2 n1i(Nout(D1)) = 0. Since the out-degree of each vertex in D1 must be at least 1, D1 is 1-regular digraph. Now, we will prove that there is no digraph Dni of order n1 > 4 such that (5.1) is zero or even positive integer. The proof will be carried out by use of the mathematical induction on the number of vertices n1 of Dni. If n1 = 4, analogously as in the case when n1 = 3, one can show that there is at least one vertex, for example uk ,in D4 such that outdegDi (uk) 4 there is at least one vertex such that (5.1) is a negative number. Let us consider a digraph Ds+1 of order s +1. By deleting an arbitrary vertex of Ds+1 we get a digraph Ds of order s, where, according to the inductive hypothesis, we can find at least one vertex, say ux, such that outdegDs (ux) < ^ nxq(Nout(Ds)) + s - 2. q=l,q=p 180 Ars Math. Contemp. 12 (2017) 111-126 If we return the removed vertex and all edges that are incident to it, we get the following inequalities: outdegDs+1 (ux) < outdegDs (ux) + 1 < s s+1 ^ nxq(Nout(Ds)) + s - 2+1 < ^ nxq(Nout(Ds+i)) + s - 1. q=1,q=i q=1,q=p Hence, according to the principle of the mathematical induction, when D2 is an isolated vertex, there is no digraph D1 of order n1 > 4 such that Nout(D) = Nout(D1VoutD2) = Q(M). □ Proposition 5.16. Let D = D1VD2 be the digraph obtained by joining two connected disjoint digraphs D1 and D2 of orders n1 and n2, respectively. If Nout(D) is the signless Laplacian matrix of some multigraph, then: 1. D1 is an isolated vertex, while D2 is any of digraphs depicted on Figure 6; 2. D1 = D2 =Ku- 3. there are no digraphs D1 and D2 of orders n1,n2 > 3 such that the statement given by the proposition holds. Figure 6: Digraphs from Proposition 5.16 Proof. Let us denote by V (D1) = {v1,v2,... ,vni} and V (D2) = {u1,u2,..., un2} the sets of vertices of digraphs D1 and D2, respectively. We have: Nout(D) = Nout(D1VD2) = Â! JT \f AT JT J A2 ) V J AT Nout(D1) + JT J A1JT + JT AT (AJJt + JT ATT )T Nout(D2) + JJT where A1 and A2 are the adjacency matrices of digraphs D1 and D2, respectively. If Nout(D) = [nij] is the signless Laplacian matrix of some multigraph, we have: n1 (1 - n2) outdegD1 (vi) + (2 - n1) n2 - nij(Nout(D1)) - m2 = 2w1, (5.2) j = 1,j=i for some non-negative integer w1 and i = 1,2,... ,n1, and n2 (1 - n1) outdegD2 (uk)+ n1 (2 - v^) - ^ nij(Nout(D2)) - ™1 = 2w2, (5.3) i=1,i=k I. M. Jovanovic: Non-negative spectrum of a digraph 181 for some non-negative integer w2 and k = 1, 2,..., n2, where m1 and m2 are the numbers of edges of digraphs D1 and D2, respectively. First, let us prove that n1 < 3. Since (5.2) means that: ni (1 - U2) outdegDl (v) > (ni - 2) n2 + ^ nj(NOMi(Di)) + m-2 j=1,j=i holds for each i = 1,2,..., n1, if we suppose that n1 > 3, we get: ni 0 > 1+ ^ nij(Nout(D1)) + m2, j=1,j=i that is a contradiction. In the same way, one can prove that n2 < 3. Statements 1. and 2. from the proposition one can get by direct analysis of (5.2) and (5.3). □ References [1] J. A. Anderson, Discrete mathematics with combinatorics, Prentice Hall, 2nd edition, 2004. [2] J. Bang-Jensen and G. Z. Gutin, Digraphs. Theory, algorithms and applications, London: Springer, 1st edition, 2000. [3] M. Behzad, G. Chartrand and L. Lesniak-Foster, Graphs and digraphs, Wadsworth International Group, 1st edition, 1981. [4] A. Berman and R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, 1979. [5] R. A. Brualdi, Spectra of digraphs, Linear Algebra Appl. 432 (2010), 2181-2213, doi:10.1016/ j.laa.2009.02.033, http://dx.doi.org/10.1016/j.laa.2009.02.033. [6] D. Cvetkovic, P. Rowlinson and S. Simic, An introduction to the theory of graph spectra, volume 75 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2010. [7] D. Cvetkovic and S. K. Simic, Towards a spectral theory of graphs based on the signless Laplacian. II, Linear Algebra Appl. 432 (2010), 2257-2272, doi:10.1016/j.laa.2009.05.020, http://dx.doi.org/10.1016/j.laa.20 0 9.05.020. [8] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra of graphs - Theory and applications, Johann Ambrosius Barth, Heidelberg, 3rd edition, 1995. [9] B. Elspas and J. Turner, Graphs with circulant adjacency matrices, J. Combinatorial Theory 9 (1970), 297-307. [10] C. Godsil and G. Royle, Algebraic graph theory, volume 207 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2001, doi:10.1007/978-1-4613-0163-9, http://dx.doi. org/10.1007/978-1-4613-0163-9. [11] J. M. Kleinberg, Authoritative sources in a hyperlinked environment, in: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 1998), ACM, New York, 1998 pp. 668-677. [12] A. N. Langville and C. D. Meyer, A survey of eigenvector methods for Web information retrieval, SIAM Rev. 47 (2005), 135-161, doi:10.1137/S0036144503424786, http://dx. doi.org/10.1137/S0036144503424786. 182 Ars Math. Contemp. 12 (2017) 183-203 [13] H. S. Ramane, H. B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog and I. Gutman, Spectra and energies of iterated line graphs of regular graphs, Appl. Math. Lett. 18 (2005), 679-682, doi:10.1016/j.aml.2004.04.012, http://dx.doi.org/10.1016/ j.aml.2004.04.012. [14] D. Randall, A. O'Neill and A. Irani, CS 6550 - Design and analysis of algorithms, Lecture and notes, 2005, http://people.math.gatech.edu/~randall/AlgsF05/ nov14.pdf. [15] D. A. Spielman, Spectral graph theory and its applications, in: FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, 2007 pp. 29-38. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 183-203 Cycle bases of reduced powers of graphs Richard H. Hammack Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, Virginia, USA Gregory D. Smith Department ofApplied Science, The College ofWilliam and Mary, Williamsburg, VA, USA Received 22 May 2015, accepted 13 September 2016, published online 26 November 2016 We define what appears to be a new construction. Given a graph G and a positive integer k, the reduced kth power of G, denoted G(k), is the configuration space in which k indistinguishable tokens are placed on the vertices of G, so that any vertex can hold up to k tokens. Two configurations are adjacent if one can be transformed to the other by moving a single token along an edge to an adjacent vertex. We present propositions related to the structural properties of reduced graph powers and, most significantly, provide a construction of minimum cycle bases of G(k). The minimum cycle basis construction is an interesting combinatorial problem that is also useful in applications involving configuration spaces. For example, if G is the state-transition graph of a Markov chain model of a stochastic automaton, the reduced power G(k) is the state-transition graph for k identical (but not necessarily independent) automata. We show how the minimum cycle basis construction of G(k) may be used to confirm that state-dependent coupling of automata does not violate the principle of microscopic reversibility, as required in physical and chemical applications. Keywords: Graph products, Markov chains, cycle spaces. Math. Subj. Class.: 05C76, 60J27 1 Introduction Time-homogenous Markov chains [19] are used as a mathematical formalism in applications as diverse as computer systems performance analysis [21], queuing theory in operations research [18], simulation and analysis of stochastic chemical kinetics [12], and biophysical modeling of ion channel gating [10]. E-mail addresses: rhammack@vcu.edu (Richard H. Hammack), greg@wm.edu (Gregory D. Smith) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract 184 Ars Math. Contemp. 12 (2017) 183-203 Many properties of a Markov chain, such its rate of mixing and its steady-state probability distribution, can be numerically calculated using its transition matrix [24]. A continuous-time Markov chain X(t) (t > 0) with a finite number of states {1,..., n} is defined by an initial probability distribution, ^¿(0) = Pr{X(0) = i}, and a transition matrix Q = (qij) where 1 < i, j < n, qij > 0 for i = j and qu = - J2j=i qij, so called because, for i = j, qij = limdi^0 Pr{X(t + dt) = j |X(t) = i}/dt. The requirement that Q has zero row sums, J2 j qij = 0, corresponds to conservation of probability, J2i ni(t) = 1, in the ordinary differential equation initial value problem, dn/dt = nQ with initial condition n(0), solved by the time-dependent discrete probability distribution n(t) = (ni(t),..., (t)) where ni(t) = Pr{X (t) = i}. A CTMC with a single communicating class of n < ro states is irreducible, positive recurrent, and has a unique steady-state probability distribution that solves nQ = 0 subject to J2i ni = 1 (by the Perron-Frobenius theorem). The Perron vector and steady-state distribution n is the limiting probability distribution of the Markov chain, ||n(t) - n || = 0, for any initial condition satisfying conservation of probability, J2i n (0) = 1. In general, the calculation of steady-state distributions and other properties for Markov chains with n states requires algorithms of O(n3) complexity. Many open questions in the physical and biological sciences involve the analysis of systems that are naturally modeled as a collection of interacting stochastic automata [3,17,23]. Unfortunately, representing a stochastic automata network as a single master Markov chain suffers from the computational limitation that the aggregate number of states is exponential in the number of components. For example, the transition matrix for k coupled stochastic automata, each of which can be represented by an v-state Markov chain, has n = vk states and requires algorithms of O(v3k) complexity. Many results are relevant to overcoming combinatorial state-space explosions of coupled stochastic automata. For example, memory-efficient numerical methods may use ordinary Kronecker representations of the master transition matrix Q = J2^ R^« where the R£n are size v, and many are identity matrices, eliminating the need to generate and store the size vk transition matrix [9]. Kronecker representations may be generalized to allow for matrix operands whose entries are functions that describe state-dependent transition rates, i.e., Q = 0«=1 Fn and Fn(i, j) : x«=1Xn ^ R where Xn(t) is the state of the nth automata [5]. Hierarchical Markovian models may be derived in an automated manner and leveraged by multi-level numerical methods [7]. Redundancy in master Markov chains for interacting stochastic automata can often be eliminated without approximation. Both lumpability at the level of individual automata and model composition have been extensively researched, though the latter reduces the state space in a manner that eliminates Kronecker structure [4,6,13]. To see this, consider k identical and indistinguishable stochastic automata, each with v states, that interact via transition rates that are functions of the global state, that is, Q = 0k F where F(i, j) : x i=1 n ^ R where n,(t) = £«=11{X„ (t) = ¿} is the number of automata in state £ As defined Q, is size vk, however, states may be lumped using symmetry in the model specification to yield an equivalent master Markov chain of size n = (k+k-1). Although model reduction in this spirit is intuitive and widely used in applications, the mathematical structure of the transition graphs resulting from such contractions does not appear to have been extensively studied. More concretely, let G represent the transition graph for an v-state automaton with transition matrix Q = (qij). As required in many applications, we assume that Q is irreducible R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 185 and that state transitions are reversible (qj > 0 ^ j > 0, i = j). Thus, the transition graph G corresponding to Q is simple (unweighted, undirected, no loops or multiple edges) and connected (by the irreducibility of Q). The transition graph G has adjacency matrix A(G) = (aj) where an = 0, and for i = j, aj = 0 when qj = 0 and aj = 1 when qij > 0. The transition graph for the master Markov chain for k automata with transition graphs Gn is the Cartesian graph product Gi DG2 □ • • • nGk. If these k automata are identical, the transition graph for the master Markov chain is the kth Cartesian power of G, that is, the k-fold product Gk = G^G^ • • • □G. The focus of this paper is the k-th reduced power of G, i.e., the transition graph of the contracted master Markov chain for k indistinguishable (but not necessarily independent) v-state automata with isomorphic transition graphs. The remainder of this paper is organized as follows. In Sections 2-3 we formally define the reduced power of a graph and interpret it as particular configuration space. Sections 46 present our primary result, the construction of minimal cycle bases of reduced graph powers. Section 7 explicates the relevance of these minimal cycle bases to applications that do not allow state-dependent coupling of automata to introduce nonequilibrium steady states. 2 Reduced Cartesian powers of a graph There are several equivalent formulations of the reduced power of a graph. For the first formulation, recall that given graphs G and H, their Cartesian product is the graph G^H whose vertex set is the Cartesian product V(G) x V(H) of the vertex sets of G and H, and whose edge set is E(G^H) = {(x,u)(y,v) | xy G E(G) and u = v, or x = y and uv G E(H)}. This product is commutative and associative [14]. For typographical efficiency we may abbreviate a vertex (x, y) of G^H as xy if there is no danger of confusion. The kth Cartesian power of a graph G is the k-fold product Gk = G^G^ • • • □G. The symmetric group Sk acts on Gk by permuting the factors. Specifically, for a permutation n G Sk the map (xi, x2, . . . , xk ) ^ (xn(1), xn(2), . . . , xw(k}) is an automorphism of Gk. The kth reduced power of G is the graph that has as vertices the orbits of this action, with two orbits being adjacent if Gk has an edge joining one orbit to the other. Said more succinctly, the reduced kth power is the quotient Gk/Sk of Gk by its Sk action. Figure 1 shows a graph G next to G2 = G^G. The S2 action on G2 has as orbits the singletons {aa}, {bb}, {cc}, {dd}, along with the pairs {ab, 6a}, {ac, ca}, {ad, da}, {bc, cb}, {bd, db}, and {cd, dc}. Let us identify a singleton orbit such as {aa} with the monomial aa = a2, and a paired orbit such as {ab, ba} with the monomial ab (with ab = ba). The reduced power G(2) appears on the right of Figure 1. Note that two monomials xy and uv are adjacent in G(2) provided that xy and uv have a common factor, and the remaining two factors are adjacent vertices of G. As each monomial xy corresponds uniquely to the 2-multiset {x, y} of vertices of G, we can also define the reduced power G(2) as follows. Its vertices are the 2-multisets of vertices of G, with two multisets being adjacent precisely if they agree in one element, and the other elements are adjacent in G. 186 Ars Math. Contemp. 12 (2017) 183-203 Qd 0 c G ad bd i cd dd ad bd ac bc 1 cc dc ac bc c2 Ob < >- 1 >— ---C > < >- i ab bb cb db ab b2 G(2) i i GOG oa S- -¿— — 3— - 3 aa ba ca da a2 cd -o d2 Figure 1: A graph G, the Cartesian square G2 = GOG, and the reduced power For each x e V(G), the vertices {xv | v e V(G)} induce a subgraph Gx = G of G(2). These subgraphs are shown dashed, dotted and solid in G(2) . Note Gx and Gy intersect precisely at vertex xy if x = y. In general, higher reduced powers G(k) can be understood as follows. Suppose V(G) = {ai, a2,..., av}. Any vertex of G(k) is the Sk-orbit of some x = (xi; x2,..., xk) G V(Gk). For each index 1 < i < v, say x has n > 0 coordinates equal to a^ Then 5^V=1 H = k, and the Sk-orbit of x consists precisely of those k-tuples in V(Gk) having Hi coordinates equal to a^ for 1 < i < v. This orbit - this vertex of G(k) - can then be identified with either the degree-k monomial an a>2 • • • a^, , or with the k-multiset { ai, ai,. .., ai | a2, a2,. .., a2 |......| av, av, ..., av }, (2.1) where v -1 dividing bars are inserted for clarity. We will mostly use the monomial notation for V (G(k)), but will also employ the multiset phrasing when convenient. Let us denote the set of monic monomials of degree k, with indeterminates V(G), as Mk(G), with M0(G) = {1}. The above, together with the definition of the Cartesian product, yields the following. Definition 2.1. For a graph G with vertex set {ai; a2,..., av}, the reduced fcth power G(k) is the graph whose vertices are the monomials an1 a22 • • • a2v G Mk (G). For edges, if aiaj is an edge of G, and f (ai; a2,... av) G Mk-i(G), then aif (ai; a2,..., av) is adjacent to ajf (ai; a2,..., av). Figure 2 shows the three-cycle G = C3 and its reduced second and third powers. Figure 3 shows the five-cycle and its reduced second and third powers. The reduced power G(k) is not to be confused with the symmetric power of G, for which each vertex represents a k-subset of V(G), and two k-subsets are joined if and only if their symmetric difference is an edge of G [1,2]. The multiset notation (2.1) gives a quick formula for the number of vertices of reduced kth powers. This presentation describes the multiset as a list of length k + v -1 involving k symbols ai, 1 < i < k, and v -1 separating bars. We can count the multisets by choosing k slots for the ai's and filling in the remaining slots with bars. Therefore, when |V(G)| = v, V (G(fc)) k + v - 1 k (2.2) R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 187 The number of vertices in Gk that are identified with vertex a^1 a^2 • • • a^" G V(G(k)) in the quotient G(k) = Gk/Sk is given by the multinomial coefficient (n n k n ) . Definition 2.1 says that for each edge a^j of G, and for each monomial f G Mk-1 (G), there is an edge of G(k) from af to aj f. Because there are ( k+m-2\ k-1 such monomials f, E(G(k)) = |E (G)| k + v - 2 k- 1 (2.3) 3 Reduced graph powers as configuration spaces The reduced power G(k) is the transition graph of the contracted master Markov chain for k identical and indistinguishable v-state automata, each with transition graph G. Consequently, an intuitive way of envisioning G(k) is to imagine it as a configuration space in which k indistinguishable tokens are placed on the vertices of G, so that any vertex can hold up to k tokens. The monomial an1 an2 • • • an" then represents the configuration in which n tokens are placed on each vertex a^ Two configurations are adjacent if one can be transformed to the other by moving a single token along an edge of G to an adjacent vertex. In this way G(k) is interpreted as the space of all such configurations. See [11] for a related construction in which no vertex can hold more than one token. The reduced power G(k) may also be interpreted as the reachability graph for a fundamental class of stochastic Petri nets with k tokens, v = |V(G)| places, and 2|E(G)| flow relations (directed arcs) between places [8,22]. The arc from place aj (origin) to place aj (destination) has firing rate n^ given by the product of transition rate qj and the number n of tokens in the origin place. That is, the aj ^ aj firing time is the minimum of n exponentially distributed random variables with expectation 1 /qj. The aj ^ aj firing rate per token will be denoted qjj [an1 an2 • • • an" ] when it is a function of the global state (token configuration) of the stochastic Petri net. The token interpretation can be helpful in deducing properties of reduced powers, such as the following. 188 Ars Math. Contemp. 12 (2017) 183-203 Proposition 3.1. The vertex a?1 a?2 • • • a?" of has degree deg (a?1 a?2 ••• a?») = (a11 a2 2 • • • aV") = > degG(ai). E i Proof. The configuration a?1 a?2 • • • can be transformed to an adjacent configuration only by moving a token on some vertex aj (with n > 1) to an adjacent vertex. □ cd2 c d Figure 3: The five-cycle C5 and its second and third reduced powers C(2) and C(3). 2 a 3 3 b e 3 3 d c R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 189 4 Cycle bases and minimum cycle bases Here we quickly review the fundamentals of cycle spaces and bases. The following is condensed from Chapter 29 of [14]. For a graph G, its edge space e(G) is the power set of E(G) viewed as a vector space over the two-element field F2 = {0,1}, where the zero vector is 0 = 0 and addition is symmetric difference. Any vector X G e(G) is viewed as the subgraph of G induced on X, so e(G) is the set of all subgraphs of G without isolated vertices. Thus E(G) is a basis for e(G), and dim(e(G)) = |E(G)|. The vertex space v(G) of G is the power set of V(G) as a vector space over F2. It is the set of all edgeless subgraphs of G and its dimension is |V(G)|. We define a linear boundary map SG : e(G) ^ v(G) by declaring that SG(xy) = x+y on the basis E(G). The subspace c(G) = ker(SG) is called the cycle space of G. It contains precisely the subgraphs in e(G) whose vertices all have even degree (that is, the Eulerian subgraphs). Because every such subgraph can be decomposed into edge-disjoint cycles, each in c(G), we see that c(G) C e(G) is spanned by the cycles in G. The dimension of c(G), denoted ^(G), is called the (first) Betti number of G. If G is connected, the rank theorem applied to SG yields P(G) = |E(G)|-|V (G)| + 1. (4.1) A basis for the cycle space is called a cycle basis. To make a cycle basis of a connected graph G, take a spanning tree T, so the set S = E(G) - E(T) has |E(G) | - |V(G) | +1 = P(G) edges. For each e G S, let Ce be the unique cycle in T + e. Then the set b = {Ce | e G S} is linearly independent. As b has cardinality P(G), it is abasis (see Figure 4). The elements of a cycle basis are naturally weighted by their number of edges. The total length of a cycle basis b is the number ¿(b) = J2cess |C|. A cycle basis with the smallest possible total length is called a minimum cycle basis, or MCB. cd 2 a 2 2 b e 2 2 d c Figure 4: A spanning tree T of G = G^2). The set S = E(G) - E(T) has P(G) = 25 - 15 + 1 = 11 edges. For each e G S, let Ce be the unique cycle in T + e. The set {Ce | e G S} is a cycle basis for G, but not a minimum cycle basis (see Figure 5). 190 Ars Math. Contemp. 12 (2017) 183-203 The cycle space is a weighted matroid where each element C has weight | C |. Hence the Greedy Algorithm [20] always terminates with an MCB: Begin with m = 0; then append shortest cycles to it, maintaining independence of m, until no further shortest cycles can be appended; then append next-shortest cycles, maintaining independence, until no further such cycles can be appended; and so on, until m is a maximal independent set. Then m is an MCB. Here is our primary criterion for determining if a cycle basis is an MCB. (See Exercise 29.4 of [14].) Proposition 4.1. A cycle basis b = {Bi; B2,..., for a graph G is an MCB if and only if every C G c (G) is a sum of basis elements whose lengths do not exceed |C |. For graphs G and H, a weak homomorphism p : G ^ H is a map p : V(G) ^ V(H) having the property that for each xy of G, either p(x)p(y) is an edge of H, or p(x) = p(y). Such a map induces a linear map p* : e(G) ^ e(H) defined on the basis E(G) as p*(xy) = p(x)p(y) provided p(x) = p(y), and p*(xy) = 0 otherwise. Similarly we define pV : v(g) ^ v(H) as pV(x) = p(x) on the basis V(G). Thus we have the following commutative diagram. (Check it on the basis E(G).) p* e(G)---> e(H) ¿G ¿H V(G)-—-> V(H) From this, p* restricts to a map c(G) ^ c(H) on cycle spaces, because if C g c(G), then ¿G (C) = 0, whence ¿Hp*(C) = pV¿G(C) = 0, meaning p*(C) G ker(£H) = c(H). Certainly if p is a graph isomorphism, then p* is a vector space isomorphism. Of special interest will be the projections pj : Gk ^ G, wherepi(xi, x2,..., xk) = x4. These are weak homomorphisms and hence induce linear maps p* : c(Gk) ^ c(G). Another important map is the natural projection n : Gk ^ G(k) sending each k-tuple x = (x i, x2,..., xk) to the monomial representing the -orbit containing x. This map n* also is a weak homomorphism, inducing a linear map n* : c(Gk) ^ c(G(k)). Lemma 4.2. If G is connected, the map n* : c(Gk) ^ c(G(k)) is surjective. Proof. Because any element of c(G(k)) is an edge-disjoint union of cycles, it suffices to show that any cycle C = /o/i • • • /„/o G c(G(k)) equals n*(C') for some C G c(Gk). For each index i, let xjyi+i G E(Gk) be an edge for which n*(xjyi+i) = n(xj)n(yi+i) = /j/i+i. (Each xj, yj is a k-tuple, and index arithmetic is modulo n.) Note that n(xj) = n(yj), meaning xj and yj are in the same -orbit, that is, yj equals xj with its coordinates permuted. We will argue that each pair yj, xj can be joined by a path Pj in Gk, with n* (Pj) = 0. This will prove the lemma because then C' = Po + xoyi + Pi + xiy2 + P2 + ... + P„ + x„yo G c(Gk) satisfies n*(C') = C. R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 191 Consider two vertices (.. .a.. .b...) and (.. .b... a...) of Gk that are identical except for the transposition of two coordinates a and b. Take a path a = v0v\ • • • vq = b from a to b in G. Now form the following two paths in Gk Q = (... a .. .b ...)(.. .vi ... b ...)(... V2 ...b...) ... (. . .b . . .b . ..) R = (...b...a...)(...b...v1 ...)(...b...v2 ...)... (...b...b...). Concatenation of Q with the reverse of R is a path from (.. .a.. .b...) to (.. .b.. .a...). Moreover n*(Q+R) = 0 because the images of the jth edges of Q and R are always equal; hence the edges cancel in pairs. As yi and xi differ only by a sequence of transpositions of their coordinates, the above construction can be used to build up a path Pi from yi to xi with n(Pi) = 0. □ We have seen that the projections pi : Gk ^ G induce linear maps c(Gk) ^ c(G). But there seems to be no obvious way of defining a projection G(k) ^ G. Still, it is possible to construct a natural linear map p* : c(G(k)) ^ c(G). To do this, recall that any edge of G(k) has form af bf where ab G E(G) and f G Mk-1(G). We begin by defining p* on the edge space. Put p*(af bf) = ab for each edge af bf in the basis E(G(k)) and extend linearly to a map p* : e(G(k)) ^ e(G). Note that J]^P* = P* ◦ n*. (Confirm it by checking it on the basis E(Gk) of e(Gk).) Now, if X G c(G(k)), then Lemma 4.2 guarantees X = n*(Y) for some Y in the cycle space of Gk. Then p*(X) = p* (n* (Y)) = Ek=ip**(Y) G c(G). We now have a linear map p* : c(G(k)) ^ c(G) for which p*(af bf) = ab. ab C5? ae_.-y ''■■.jab ¿bey- ad d2 cd b2e2 ae y'' '■■. ab be ",bc de\" o cd c2 be bd ?bc cd a 2 a Figure 5: The union {G^a} U b is an MCB for c(g(2)) = c(C5 a) Q s(g(2)). 192 Ars Math. Contemp. 12 (2017) 183-203 5 Decomposing the cycle space of a reduced power This section explains how to decompose the cycle space of a reduced power into the direct sum of particularly simple subspaces. To begin, notice that if f is a fixed monomial in Mk-1 (G), then there is an embedding G ^ G(k) defined as x ^ xf. Let us call the image of this map Gf. Notice that Gf is an induced subgraph of G(k) and is isomorphic to G. Proposition 5.1. For any fixed f e Mk-i(G), we have C (G(k)) = C (Gf) 0 ker(p*). Proof. Consider the map p* : C(G(k)) ^ C(G). Its restriction C(Gf) ^ C(G) is a vector space isomorphism. The proof now follows from elementary linear algebra. □ Next we define a special type of cycle in a reduced power. Given distinct edges ab and cd of G and any f e Mk-2 (G), we have a square in G(k) with vertices acf, bcf, bdf, adf. Let us call such a square a Cartesian square, and denote it as (abUcd)f. See Figure 6. We regard this as a cycle in the cycle space; it is the subgraph of G(k) that is precisely the sum of edges acf bcf + bcf bdf + bdf adf + adf acf. (Observe that this sum is zero if and only if ab = cd.) We remark that although a subgraph Gf may have squares, they are not Cartesian squares because they do not have the form specified above. Define the square space s(G(k)) to be the subspace of c(G(k)) that is spanned by the Cartesian squares. If S is a Cartesian square, then p*(S) = 0, so s(G(k)) C ker(p*). In the remainder of the paper we will show that in fact s(G(k)) = ker(p*), so that Proposition 5.1 gives c(G(k)) = c(Gf) 0 s(G(k)). Simultaneously we will craft a simple MCB for G(k) by concatenating MCBs of c(Gf) and s(G(k)). See Figure 5 for an example. 6 Cycle bases for reduced powers This section describes a simple cycle basis for the reduced kth power of a graph G. If G has no triangles, this cycle basis will be an MCB. (We do not consider MCBs in the cases that G has triangles because the applications we have in mind do not involve such situations. Constructing MCBs when G has triangles would be an interesting research problem.) Let G be a connected graph with v vertices and e edges. Recall that by Equations (2.2) and (2.3), the graph G(k) has (k+v-1) vertices, identified with the monomials Mk (G), and e(k+V-2) edges. Thus any cycle basis has dimension We first examine the square space. Any pair of distinct edges ab and cd of G corresponds to a Cartesian square (abDcd)f, where f e Mfc_2(G), so there are (2)(fc+V23) adf q- 2. (6.1) R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 193 such squares. But this set of squares may not be independent. Our first task will be to construct a linearly independent set of Cartesian squares. To begin, put V(G) = {a1, a2,..., av}. Let T be a rooted spanning tree of G with root a1, and arrange the indexing so its order respects a breadth-first traversal of T, that is, for each i the vertex ai is not closer to the root than any aj for which j < i (see Figure 7). 05 Figure 7: A rooted spanning tree T of G with V(G) = {a1, a2,..., av}, root a1, and indexing that respects a breadth-first traversal of T. With this labeling, any edge of T is uniquely determined by its endpoint aj that is furthest from the root. For each 2 < i < v, let ej be the edge of T that has endpoints aj and aj, with aj further from the root than ai. Let Mk-2(a1, a2,.. .aj) denote the monic monomials of degree k — 2 in indeterminates a1,a2,... ,aj, with 1 < j < v. Define the following sets of Cartesian squares in G(k). T = {(ejUej )f | 2 < i 3 and take three distinct edges aiaj-, a^am and apaq in G, and let f € Mk-3(G). Figure 8 indicates that these edges result in a cube in the kth reduced power. Each of the six square faces of this cube is in the square space. But the faces are dependent because any one of them is a sum of the others. Call a square face such as (aiaj Dagam)aq f a "top square" of a cube if the monomial aq f involves an indeterminate at with t > max{i, j, i, m}. Sets T and Q are constructed so as to contain no top squares. 194 Ars Math. Contemp. 12 (2017) 183-203 (A configuration of the type illustrated in Figure 8 may not always be a cube in the combinatorial sense. The reader is cautioned that if a»aj, a£am and apaq are the edges of a triangle in G, then two of the diagonally opposite vertices of the "cube" are the same, as (3) in K3 , shown in Figure 2. Here there is only one cube, which takes the form of a central vertex connected to the six vertices of a hexagon. This will cause no difficulties in what follows, even if we entertain the possibility that G does indeed have triangles.) There is another kind of dependency that is ruled out in the definition of T and Q, and we now sketch it. First, imagine G2. Consider two cycles A and B in G each having exactly one edge not in T, say a»aj and a^am, respectively. Envision AdB is as a torus in G2 with square faces, each edge shared by two faces. In adding up all the faces, the edges cancel in pairs, giving 0, so the squares are dependent. Removing the face a»aj da^am removes the dependency. Such squares a^ajOaeam show up in G(2)/ C G(k) as squares (ajajDajam)/ with a»aj, apaq G E(G) - E(T). Sets T and Q contain no such squares. Proposition 6.1. The set B = T U Q is linearly independent. Proof. We first show that T is linearly independent. Let X = J2(eiaej)/« be a sum of elements of T. Form the forest F C T consisting of all edges e» and ej that appear as edges of a squares in this sum, and let ab be an edge of F for which b is a leaf. Then any term (a^am Dab)/n of the sum is the unique square in the sum containing the edge a^b/n amb/n. Because no term can cancel this edge, we get X = 0, so T is linearly independent. To see that Q is linearly independent, consider a sum X = J2(aeamdej)/n of squares in Q. Again form a forest F C T of the edges ej and let ab be as before. Then any term (a£amDab)/„ is the unique square in the sum containing the edge a^b/« amb/n. Then X = 0 because no other term in the sum can cancel this edge; hence Q is linearly independent. Now we argue that the spans of T and Q have zero intersection. By the previous paragraph, any nonzero linear combination of squares in Q has edges of form (a£amDab)/„, with a^am G E(G) - E(T). But no linear combination of squares in T has such edges. Hence the spans have zero intersection, so b is linearly independent. □ Our next task is to show that b is actually a basis for the square space. In fact, we will show more: it is also a basis for ker(p*), and s(G(k)) = ker(p*). Our dimension counts will involve finding |T| and |Q|, and for this we use the following formulas. The first is standard; both are easily verified with induction. c) + c:') + c:2) + ••• + (D = ('+;+') («.2) oQ + i(r:')+2(rf) +... + <:■) = c::1)-r:::1) («.3) Take an edge ej of T with 3 < j. From its definition, T has (j - 2) (k+j_-23) squares of form (ejdej)/. We reckon as follows, using Equations (6.2) and (6.3) as appropriate. R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 195 |Y| = P - + --3) »; - 2)(k+j-V j=3 (j - 2)fk i = i V k - ^ = t - i)fk+--3) -t fk + --3) +1 j=1 v y j=1 v y . fk + v - 2) fk + v - 2) fk + v - 2) = (V - :)f k - 1 ) - f k ) H k - 1 J+1 , ,,fk + v - 2\ fk + v - A = (v -^ k -1 H k J+1- (6.4) Now, given and edge ej of T with 2 < j, the set Q has P(G)(squares of form (a,£amOej)f. Consequently n = ^(G) ¿c+--3) 3=2 V 7 = *G) - r) -1 = £(G)(k + -- ^(G)- (6.5) Proposition 6.2. The set B = Y U Q is a basis for the square space of the reduced kth power of G. Moreover, the square space equals ker(p*). Proof. By Proposition 6.1, the set b is linearly independent; and it is a subset of the square space by construction. We saw earlier that the square space is a subspace ker(p* ). To finish the proof we show that ker(p*) has dimension |b|. By the rank theorem applied to the surjective mapp* : c(G(k)) ^ c(G) we have dimker(^*) = fi(G(2)) - fi(G). This with Equations (6.1), (6.4) and (6.5), as well as the fact that (v - 1) + fi (G) = e, gives |b| = |Y| + |Q| = (v - 1>(k + --2) - (k + V - 1) +1 + «< + --2) -fi(G) = e(k + -- 2) - (k + k - 1) +1 - ^(G) = fi(G(2)) - fi(G) = dimker(p*). Therefore b is a basis for both s(G(k) ) and ker(p* ). □ If k = 2, then b = {abOcd | ab, cd e E(G)} - {abOcd | ab, cd e E(G) - E(T)}, so |b| = (2) - (W2g)). It is interesting to note that if fi(G) < 1, then (^2G)) =0 and b consists of all squares in the square space; in all other cases it has fewer squares. 196 Ars Math. Contemp. 12 (2017) 183-203 a\ Figure 9: With T as indicated, the sets of squares Y and Q form a basis b = Y U Q of the square space of C5;3). Here Y = {(abdbc)f | / G {a, b, c}} U {(abdcd)f, (bcdcd)f | / G {a, b, c, d}} U{(abdde)f, (bcdde)f, (cddde)f | / G {a, b, c, d, e}}. Also Q = {(aeDab)f | f G {a, b}} U {(aeDbc)f | f G {a, b, c}} U {(aeDcd)f | f G {a, b, c, d}} U {(aeDde)f | f G {a, b, c, d, e}}. Note |Y| = 24 and |Q| = 14. The square (abDcd)e G b is the "top square" of the Cartesian cube abDcdnde. We now can establish the main result of this section, namely a construction of an MCB for the reduced kth power. Take an f G Mfc_i (G). Propositions 5.1 and 6.2 say c(G(k)) = c(Gf) 0 s(G(k)). (6.6) To any cycle C = c1 c2... cn in G, there corresponds cycle Cf = c1f c2f ... cnf in G(k). Theorem 6.3. Take a cycle basis C = {C1, C2,..., C^(G)} for G, and let B be the basis for s (G(k)) constructed above. Fix f G Mk-1(G) and put cf = {C1f, C2f,..., c^(g)/}. Then cf U b is a cycle basis for If C is an MCB for G, and G has no triangles, then this basis is an MCB for G(k). Proof. That this is a cycle basis follows immediately from Equation (6.6). R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 197 Now suppose c is an MCB for G, and that G has no triangles. It is immediate that G(k) has no triangles either. The proof is finished by applying Proposition 4.1. Take any C e c(G(k)), and write it as C = E Gi + E Bj, iei jeJ where the Gi are from cf and the Bj are from b. According to Proposition 4.1, it suffices to show that C has at least as many edges as any term in this sum. Certainly C is not shorter than any square Bj (by the triangle-free assumption). To see that it is not shorter than any Gi in the sum, apply p* to the above equation to get P*(C) = EP*(Gi) • iei Because p* : c(Gf) ^ c(G) is an isomorphism, the terms p*(Gi) are part of an MCB for G, and thus |p*(C)| > |p*(Gi)| = |Gi| for each i, by Proposition 4.1. Also |C| > |p*(C)| (as some edges may cancel in the projection) so |C| > |Gi|. □ Although Theorem 6.3 gives a simple MCB for reduced powers of a graph that has no triangles, the constructed basis is definitely not minimum if triangles are present. Several different phenomena account for this. Consider the case k = 2. First, if G has triangles, then for each vertex x of G, the second reduced power contains a copy Gx of G. These copies are pairwise edge-disjoint; an MCB would have to capitalize on triangles in each of these copies at the expense of squares in the square space. Moreover, as Figure 2 demonstrates, some of the squares in the square space will actually be sums of two triangles. The figure also shows that for a triangle A = abc in G, we do not get just the three triangles Aa, Ab and Ac, but also a fourth triangle ab bc ca not belonging to any Gx. We do not delve into this problem here. 7 Discussion We have defined what appears to be a new construction, the kth reduced power of a graph, G(k), and have presented a theorem for construction of minimal cycle bases of G(k). When G is the transition graph for a Markov chain, G(k) is the transition graph for the configuration space of k identical and indistinguishable v-state automata with transition graph G. Symmetry of model composition allows for interactions among stochastic automata, so long as the transition rates qij for i, j e {1, 2, • • • , v}, i = j are constant or functions of the number of automata n(t) in each state, 0 < n^(t) < k, 1 < I < v. G(k) does not pertain if transition rates depend on the state of any particular automaton, Xn(t) e {1, 2, ••• , v}, n e {1,2, ••• ,k},as this violates indistinguishability. For concreteness, consider a stochastic automata network composed of three identical automata, each with transition graph C5 and generator matrix, ( o qab [•] 0 0 qae \ Q = qba o qbc 0 0 0 qcb o qcd 0 0 0 qdc o qde V qea 0 0 qed o (7.1) 198 Ars Math. Contemp. 12 (2017) 183-203 where o's indicate the values required for zero row sum, qu = — J2j=i qj < 0, and qab[• ] indicates a functional transition rate that depends on the global state of the three automata. Assume constant transition rates qbc = qcd = qde = qea = ^ > 0 and qba = qcb = qdc = qed = qae = v > 0. Further assume that the automata may influence one another through the state-dependent transition rate, qab[ • ] = A + a (na[• ] - 1) + pub[• ] + 7«c[• ] + Jnd[• ] + ene[ • ], (7.2) where a, P, 7, J, e > 0 and [ • ] denotes the global state a11 a??2 • • • aV" that is the functional transition rate's argument. The transition rate qab : Mk(a1; a2,•• • , av) ^ R is a function of the global state via n : Mk(a1; a2,•• • , av) ^ N defined by n[ap a^2 • • • aV"] = The three automata are uncoupled when a, P, 7, J, e = 0 because this eliminates the dependence of qab[ • ] on the global state. (In this model specification, coupling an isolated component automaton to itself is equivalent to absence of coupling. Because qab[ • ] is the rate of an a ^ b transition, qab[ • ] is only relevant when the isolated automaton is in state a. This functional transition rate has the property that qab[a] = A when a, P, 7, J, e > 0 because nx[y] = 1 for x = y and 0 otherwise.) The transition matrix for the master Markov chain Q(3) is defined by the model specification in the previous paragraph. For example, the transition rate from global state ad2 to global state abd is q(3)[ad2, abd] = because nd[ad2] = 2 and qdb = ^ is not a function of the global state. Other examples are q(3)[c3, c2d] = nc[c3]qcd = 3v, q[a2c, a2d] = nc[a2c]qcd = v, q(3) [abe, b2e] = na[abe]qab[abe] = A + a (na [abe] — 1) + P n [abe] + 7 nc [abe] + J n^ [abe] + e ne [abe] = A + P + e q(3)[a3,a2b] = na[a3]qab[a3 ] = 3 (A + a (na[a3] — 1) + Pnb[a3] + 7n^a3] + Jnd[a3] + ene[a3]) = 3(A + 2a) q(3) [a2c, abc] = na[a2c]qab[a2 c] = 2 (A + a (na[a2c] — 1) + Pnb[a2c] + 7nc[a2c] + Jn^[a2c] + ene[a2c]) = 2 (A + a + 7) . This process of unpacking the model specification yields a master Markov chain with n = (fc+V-1) = (3+53-1) = 35 states. The master Markov chain has 210 transition rates qij > 0 corresponding (in pairs) to the 5(3+5-2) = 105 edges of the master transition graph C^3). The construction of minimal cycle bases of G(k) provided by Theorem 6.3 is especially relevant to stochastic automata networks that arise in physical chemistry and biophysics [15]. For many applications in these domains, the principle of microscopic reversibility requires that the stationary distribution of uncoupled automata satisfying global balance, nQ = 0 subject to J2i n = 1, also satisfies a stronger condition known as detailed balance, qij = qjinj. i=j j=i R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 199 ni[imf ]qij [imf ] imf n[Uf ]qe [if ] nj [jmf ]qji[jmf ] nm [imf] qmi [imf] n [jtf] qim [jtf ] ni[i£f ]qij [itf ] jmf nm [jmf ]qmi[jmf ] if jtf nj jtf ]qji[jf ] Figure 10: Many cycles of the directed, weighted transition graph for a master Markov chain for k coupled v-state automata correspond to Cartesian squares (ijaim)f of the minimal cycle basis for the undirected, unweighted transition graph G(k), where i, j, £,m e {ai, a,2, ••• , av } and f e Mk~2(ai,a,2, ••• , av). In other words, nonequilibrium steady states are forbidden. Markov chains have this property when the transition rates satisfy the Kolmogorov criterion, namely, equality of the product of rate constants in both directions around any cycle in the transition matrix Q [16]. For an isolated automaton with transition graph C5 and transition matrix (7.1), the Komol-ogorov criterion is qab[a] qbc qcd qde qea = qae qed qdc qcb qba■ (7.3) Substituting the transition rates of the model specification, both those that are constant as well as qab [a] = A (7.2), yields the following condition on model parameters, X,m4 = v5, (7.4) that ensures the stationary distribution of an isolated automaton will satisfy detailed balance. (3) By constructing the minimal cycle basis of C5 , we may verify that the master Markov chain for three uncoupled automata, each with transition graph C5, also exhibits microscopic reversibility under the same parameter constraints. To see this, recall that the minimal cycle basis of C53) has 39 linearly independent cycles. Microscopic reversibility for the master Markov chain for three uncoupled automata requires that, given (7.4) and a, /3, = 0, 39 Komolgorov criteria are satisfied, each corresponding to a Ci in the MCB for c(3). One cycle in the MCB for C(3) takes the form C5 f for fixed f e M2(a, b, c, d, e). The Kolmogorov criterion for this cycle is na[af]qab[af] • nb[bf]qbc[bf] • nc[cf]qcd[cf] • nd[df]qde[df] • ne[ef]qea[ef] = na [af ]qae[af ] • ne[ef ]qed[ef ] • nd[df ]qdc[df ] • nc[cf ]qcb[cf ] • nb[bf ]qba[bf ], where, for typographical efficiency, here and below, we drop the superscripted (3) on the 200 Ars Math. Contemp. 12 (2017) 183-203 transition rates q(3) [•, •] of Q(3). Canceling identical terms of the form nx[xf] gives qab[af] • qbc[bf] • qcd[cf] • qde[df] • qea[ef] = qae[af] • qed[ef] • qdc[df] • qcb[cf] • qba[bf}. When this expression is evaluated, the result is another instance of (7.4), which is satisfied by assumption. (3) The remaining 38 Ci in the MCB for Cg ' are Cartesian squares (see Figure 10) that yield Kolmogorov criteria of the form, nj [imf ]qjj [imf] • nm[jmf]qm£[jmf] • nj [j£f\qji[j£f\ • ne[i£f]qem[i£f] = nm[imf]qm£[imf] • nj [i£f]qij [i£f] • ne[j£f]qem[j£f] • nj [jmf]qji[jmf], where f e Mi(a, b, c, d, e). For x = y, nx[xyf] = nx[x] + nx[y] + nx[f] = 1 + nx[f], so this criterion simplifies to (1 + nj[f])qjj [imf] • (1 + nm[f])qm£[jmf] • (1 + nj [f])qji[j£f] • (1 + ne[f])qem[i£f] = (1 + nm[f])qm£[imf] • (1 + ni[f])qj [i£f] • (1 + ne [f])qem[j£f] • (1 + nj [f])qji[jmf]. Canceling identical terms of the form (1 + nx[f ]) gives qij [imf ] qm [jmf] qji [j£f ] qim [i£f ] = qmi [imf ] qj [i£f ] qim [j£f ] qji [jmf ] (7.5) for (ijD£m)f e B = Y U Q with f e Ml(al,a2,... ,av). When the automata are not coupled, a, /3, = 0, the transition rates are not functions of the global state, and every factor on the left hand side has an equal partner on the right. Consequently, the 38 squares of b correspond to cycles in Q(3) that satisfy Komolgorov criteria. (3) We have shown that every cycle in the MCB for Cg ', given by Cg a U b, corresponds to a cycle in Q(3) that satisfies a Komolgorov criterion. For every cycle in Q(3), there is a representative in the cycle space c(C(3)) that is a linear combination (over the field F2) of elements of the MCB. It follows that every cycle in the master Markov chain satisfies the Komolgorov criterion. Thus, we conclude that the master Markov chain for three uncoupled automata exhibits microscopic reversibility provided an isolated automaton has this property. This property is expected, and yet important for model verification. In many applications, it is important to establish whether or not model composition (i.e., the process of coupling the automata) results in a master Markov chain with nonequilibrium steady states, in spite of the fact that an isolated component automaton satisfies detailed balance. Such nonequilibrium steady states may be objects of study or, alternatively, the question may be relevant because the master Markov chain is not physically meaningful when model composition introduces the possibility of nonequilibrium steady states [15]. Our construction of minimal cycle bases of reduced graph powers provides conditions sufficient to ensure that model composition does not introduce nonequilibrium steady states. In general, it is sufficient that (7.5) hold of every Cartesian square (ijU£m)f of the MCB for the undirected, unweighted transition graph G(k). In the example under discussion, many of these Komolgorov criteria do not involve the functional transition rate qab[}\ these conditions are satisfied without placing any constraints on the coupling parameters a,y,S,c. The remaining constraints take the form qab[amf ] qm£ [bmf ] qba [b£f ] qem [a£f ] = qm£ [amf ] qab[a£f ] qem [b£f ] qba [bmf ] (7.6) R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 201 for ¿m G {bc, cd, de, ae}. The Cartesian squares of concern are elements of the set {(ab^m)/ | ¿m G {bc, cd, de}} C T and (ae^ab)/ G Q. Note that ¿m = ab and, consequently, qm£ [bm/ ] = qm£ [am/], q^K/] = q£m[b/] and q6o[b/] = q6o[bm/] = v. Thus, (7.6) simplifies to qab[a/j = qab[am/j ¿m G {bc, cd, de, ae}. (7.7) To see how this requirement constrains the coupling parameters a, 7, S, e, we expand both sides of (7.7) using (7.2), for example, qab[a/j = A + a(na[a/] - 1) + ¡n6[a^/j + 7nc[a/] + Snd[a/] + ene[af/] = A + an« [/ ] + £n6[/] + 7nc[/] + Sn„[/] + ene[^/] where we used na[a/] = 1 + n«/]. Subtracting both sides of (7.7) by A + ana[/] + ¡™b[/] + 7nc[/] + Snd[/] + ene[/] and using n-J/] = n^] + nx[/] we obtain an« [¿] + ¡n-b [¿]+7™c [¿] + Snd [¿]+ene [¿] = an« [m] + ¡n [m] + ■7nc [m] + Snd [m] + ene [m] for ¿m G {bc, cd, de, ae}. These four equations yield four parameter constraints that ensure detailed balance in the master Markov chain for the three coupled stochastic automata, for example, ¿m = bc gives an« [b] + [b] + 7nc [b] + Snd [b] + en [b] = ana [c] + [c] + 7nc [c] + Snd [c] + e«e [c], which implies that ft = 7. Substituting ¿m = cd, de and ae, we find 7 = S, S = e and a = e, respectively. We conclude that a = ft = 7 = S = e. In our example, the three automata are coupled when one or more of a, 7, S, e is (3) positive. The analysis of Cartesian squares in the MCB for Cg shows that coupling the three automata in the manner specified by (7.2) will introduce nonequilibrium steady states unless the coupling parameters are equal. This result is intuitive because J2i n [•] = k = 3 and, consequently, equal coupling parameters a = ft = 7 = S = e correspond to a functional transition rate that, for every global state, evaluates to the constant qa6[ ] = A + a(k - 1) = A + 2a. The simplicity of this parameter constraint is a consequence of evaluating (7.5) in the context of the example model specification. In general, the resulting constraints may be more complex and less restrictive. Any choice of model parameters that simultaneously satisfies qij [im/] qmi[jm/] qji[j/] q*m[i/] = qm^im/] qj [i/] qim[j/] qji[jm/] for (ij□¿m)/ G B = Y U Q with / G Mk-2(a1, a2,..., av) are conditions sufficient to ensure that the process of model composition (i.e., coupling k identical and indistinguishable v-state automata) does not introduce a violation of microscopic reversibility. Acknowledgments The work was supported by National Science Foundation Grant DMS 1121606. GDS acknowledges a number of stimulating conversations with William & Mary students enrolled in Spring 2015 Mathematical Physiology and Professor Peter Kemper. Thanks also to the referee for many helpful suggestions. 202 Ars Math. Contemp. 12 (2017) 183-203 References [1] A. Alzaga, R. Iglesias and R. 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Klavzar, Handbook of product graphs, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2nd edition, 2011. [15] T. Hill, Free Energy Transduction in Biology and Biochemical Cycle Kinetics, Springer-Verlag, New York, 1989. [16] F. Kelly, Reversibility and Stochastic Networks, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2011. [17] T. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, volume 324 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1999. [18] M. Neuts, Structured stochastic matrices of M/G/1 type and their applications, volume 5 of Probability: Pure and Applied, Marcel Dekker, Inc., New York, 2nd edition, 1989. [19] J. Norris, Markov chains, Cambridge University Press, Cambridge, 1997. [20] J. G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. [21] B. Plateau and J. Fourneau, A methodology for solving markov models of parallel systems, J. Parallel Distrib. Comput. 12 (1991), 370-387. R. H. Hammack and G. D. Smith: Cycle bases of reduced powers of graphs 203 [22] W. Reisig, Understanding Petri Nets: Modeling Techniques, Analysis Methods, Case Studies, Springer-Verlag, Berlin, 2013. [23] W. Richoux and G. Verghese, A generalized influence model for networked stochastic automata, IEEE Trans. Syst., Man, and Cybern. A, Syst., Humans 41 (2011), 10-23. [24] W. Stewart, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, 1994. ARS MATHEMATICA CONTEMPORANEA Author Guidelines Before submission Papers should be written in English, prepared in LTEX, and must be submitted as a PDF file. The title page of the submissions must contain: • Title. The title must be concise and informative. • Author names and affiliations. For each author add his/her affiliation which should include the full postal address and the country name. If avilable, specify the e-mail address of each author. Clearly indicate who is the corresponding author of the paper. • Abstract. A concise abstract is required. The abstract should state the problem studied and the principal results proven. • Keywords. Please specify 2 to 6 keywords separated by commas. • Mathematics Subject Classification. Include one or more Math. Subj. Class. codes - see http://www.ams.org/msc. After acceptance Articles which are accepted for publication must be prepared in LTEX using class file amcjoucc.cls (which has replaced the deprecated amcjou.cls class file) and the bst file am-cjoucc.bst (if you use BibTEX). If you don't use BibTEX, please make sure that all your references are carefully formatted following the examples provided in the sample file. 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Subscription Order Form Name: .............................. E-mail: .............................. Postal Address: ...................... I would like to subscribe to receive ...... copies of each issue of Ars Mathematica Contemporanea in the year 2017. I want to renew the order for each subsequent year if not cancelled by e-mail: □ Yes □ No Signature: Please send the order by mail, by fax or by e-mail. By mail: Ars Mathematica Contemporanea UP FAMNIT Glagoljaska 8 SI-6000 Koper Slovenia By fax: +386 5 611 75 71 By e-mail: info@famnit.upr.si viii ars mathematica contemporanea Dan Archdeacon (11 May 1954 to 18 February 2015) Professor Dan Archdeacon was a world-renowned mathematician, an intellectually engaging teacher, and a highly respected colleague. His research interests were in graph theory, combinatorics, and theoretical computer science. He published over seventy refereed papers in these fields, a majority of them in topological graph theory. His dissertation entitled "A Kuratowski Theorem for the Projective Plane" contains a proof of a ground-breaking and highly cited theorem which gives the extension of the Ku-ratowski theorem for the projective plane. This result has not been superseded. His service to the mathematical community is widely appreciated. For over a decade, he was an editor of the Journal of Combinatorial Theory, Series B, and then he managed the offices of Journal of Graph Theory as its managing editor. He also served as a referee to over 30 journals in his field, and with Jeff Dinitz, he organised seven workshops in the Vermont Summer School on Combinatorics and Graph Theory. He started and maintained an online compendium of open problems in topological graph theory, which was an inspiration to several generations of the researchers in the field. For most of his career, Dan taught at the University of Vermont, where he was named a University Scholar for the 2003/04 academic year. He was a Fulbright Teaching Fellow at the Riga Commerce School (Latvia), and visiting professor at the University of Auckland (New Zealand), Yokohama National University (Japan), the Technical University of Denmark, and the Open University (UK). He was an invited speaker at mathematics conferences across the globe. His talks were entertaining and always well attended. Some of us had the privilege of working with Dan for more than two decades, on projects that resulted in many joint papers. Doing research with Dan was always a fantastic experience, both mathematically and socially. Dan was a very quick and sharp thinker. When tackling a research problem he quickly plunged into the absolute depth of the matter, but kept thinking in terms of a bigger picture of the situation at all stages. He was a true visionary, outlining avenues of research followed by others. At the same time he was a person with a great sense of humour, highly intelligent and very pleasant. He will be missed by an entire generation of scholars. Dan visited Slovenia twice. In June 1999, he participated at the 4th Slovenian International Graph Theory Conference at Lake Bled, speaking of the representativity of planar graphs. During his second visit, in January 2014, he was already fighting cancer, but with the same optimism that he showed in his energy and passion for research. He spoke about toroidal triangulations being geometric, and initiated a problem on generalisation of Gauss words, which led to his last paper, published in this issue of Ars Mathematica Contemporanea. Drago Bokal, Bojan Mohar, Jozef Siran ix ars mathematica contemporanea 7nmrtt«> I Í/NIVÍ*1 7th PhD Summer School in Discrete Mathematics Rogla, Slovenia, July 23 - July 29, 2017 https://conferences.famnit.upr.si/event/2/ Summer School Programme: Aimed at bringing PhD students to several open problems in the active research areas, two minicourses (10 hours of lectures each) will be given on the following topics: • Theory of pseudo-Boolean functions and binary optimization, Endre Boros, (MSIS and RUTCOR, Rutgers University, New Jersey, USA) • A hundred years of Graph theory, Robin Wilson, (Open University, London, UK) Venue: Rogla is a highland in the north-eastern part of Slovenija, located 130 km by road from Slovenian capital Ljubljana. At around 1500m above sea level, the beautiful natural scenery of Rogla provides pleasant climate conditions and stimulating working environment. Organized by University of Primorska, UP IAM and UP FAMNIT, in collaboration with Centre for Discrete Mathematics UL PeF and Slovenian Society for Discrete and Applied Mathematics. Scientific Committee: K. Kutnar, A. Malnic, D. Marusic, S. Miklavic, T. Pisanski, P. Sparl, B. Zgrablic. Organizing Committee: B. Frelih, A. Hujdurovic, B. Kuzman, R. PoZar. Sponsored by Slovenian National Research Agency (ARRS) and Ministry of Education, Science and Sport (MIZS). For more information, visit our website or email your inquiry to sygn@upr.si. x ars mathematica contemporanea Petra Sparl Award 2018: Call for nominations The Petra Sparl Award has been established to recognise (in each even-numbered year) the best paper published recently by a young woman mathematician in one of the two journals Ars Mathematica Contemporanea (AMC) and The Art of Discrete and Applied Mathematics (ADAM). The award is named in memory of Dr Petra Sparl, a talented woman mathematician with a promising future who worked in graph theory and combinatorics, but died mid-career in 2016 after a battle with cancer. This award consists of a certificate with the recipient's name, and an invitation to give a lecture at the Mathematics Colloquium at the University of Primorska, and to give lectures at the University of Maribor and University of Ljubljana. The Petra Sparl Award Committee is now calling for nominations for the first award. Eligibility: Each nominee must be a woman author or co-author of a paper published either in AMC or ADAM in the last five years, who was at most 40 years old at the time of the paper's first submission. Nomination format: Each nomination should specify the following: (a) the name, birth-date and affiliation of the candidate; (b) the title and other bibliographic details of the paper for which the award is recommended; (c) reasons why the candidate's contribution to the paper is worthy of the award, in at most 500 words; and (d) names and email addresses of one or two referees who could be consulted with regard to the quality of the paper. Procedure: Nominations should be submitted by email to any one of the three members of the Petra Sparl Award Committee (see below), by 31 August 2017. Award Committee: Marston Conder: m.conder@auckland.ac.nz Asia Ivic Weiss: weiss@mathstat.yorku.ca Aleksander Malnic: Aleksander.Malnic@guest.arnes.si xi Printed in Slovenia by Tiskarna Kostomaj d.o.o.