The ADAM graph and its configuration1 It is well-known that exactly seven of the generalised Petersen graphs are symmetric (= arc- transitive), namely the following: • G(4, 1) – the cube graph, • G(5, 2) – the Petersen graph, • G(8, 3) – the Möbius-Kantor graph, • G(10, 2) – the dodecahedron graph, • G(10, 3) – the Desargues graph, • G(12, 5) – the Nauru graph, and • G(24, 5) – the graph that we hereby name the ADAM graph. Both G(8, 3) and G(10, 3) are associated with point-line configurations: G(8, 3) is the Levi graph (= incidence graph) of the Möbius-Kantor (83) configuration, while G(10, 3) is the Levi graph if the Desargues (103) configuration. A point-circle configuration is called an isometric configuration if all circles have the same radius, and a graph drawn in the plane is called unit-distance graph if all straight edges have the same length. The above figures depict an isometric point-circle configuration (243) on the left, whose Levi graph is the generalised Petersen graph G(24, 5) drawn as the unit-distance graph on the right. The central detail has been adopted as the logo of our new journal, The Art of Discrete and Applied Mathematics, and because its abbreviation is ADAM, we propose that the generalised Petersen graph G(24, 5) and the corresponding (243) configuration be called respectively the ADAM graph and the ADAM configuration. Dragan Marušič and Tomaž Pisanski Editors In Chief 1We would like to thank Nino Bašić and Arjana Žitnik for drawing both figures. i ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.01 https://doi.org/10.26493/2590-9770.1216.6d2 (Also available at http://adam-journal.eu) The k-independence number of graph products⇤ Yaping Mao School of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai 810008, China Eddie Cheng Department of Mathematics and Statistics, Oakland University, Rochester, MI USA 48309 Zhao Wang † School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China Zhiwei Guo School of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai 810008, China Received 13 November 2016, accepted 3 April 2017, published online 27 June 2017 Abstract The concept of k-independent number is a natural generalization of classical inde- pendence number. A k-independent set is a set of vertices whose induced subgraph has maximum degree at most k. The k-independence number of G, denoted by ↵k(G), is de- fined as the maximum cardinality of a k-independent set of G. In this paper, we study the k-independence number on the lexicographical, strong, Cartesian and direct product and present several upper and lower bounds for these products of graphs. Keywords: Independence number, k-independent set, k-independence number, lexicographical prod- uct, strong product, Cartesian product, direct product. Math. Subj. Class.: 05C69, 05C76 ⇤Supported by the National Science Foundation of China (Nos. 11601254, 11551001, 11371205, 11661068, 11161037, 11101232, and 11461054) and the Science Found of Qinghai Province (Nos. 2016-ZJ-948Q, and 2014-ZJ-907). The authors are very grateful to the referees for their valuable comments and suggestions, which greatly improved the presentation of this paper. †Corresponding author. E-mail addresses: maoyaping@ymail.com (Yaping Mao), echeng@oakland.edu (Eddie Cheng), wangzhao@mail.bnu.edu.cn (Zhao Wang), guozhiweic@yahoo.com (Zhiwei Guo) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.01 1 Introduction Graphs considered in this paper are undirected, finite and simple. We refer to [1] for un- defined notations and terminology. In particular, we use (G) and (G) to denote the maximum degree and minimum degree of a graph G, respectively. If X ✓ V (G) or X ✓ E(G), then G[X] is the subgraph of G induced by X . For two subsets X and Y of V (G) we denote by EG[X,Y ] the set of edges of G with one end in X and the other end in Y . Independence number is one of the most basic concepts in graph theory. A subset S ✓ V (G) is said to be independent if E(G[S]) = ;. The independence number of G denoted by ↵(G) is the size of a maximum independent set in G. In [6, 7], Fink and Jacobson generalized the concept of independent set. In this paper, k will be an integer. We say that a subset S of V is k-independent if (G[S])  k, that is, the maximum degree of the subgraph induced by the vertices of S is less or equal to k. The k-independence number, denoted ↵k(G), as the maximum cardinality of a k-independent set. Thus for k = 0, the 0-independent is the classical independent set. Every k-independent set is (k + 1)-independent; so ↵k+1(G) ↵k(G) for a graph G. Moreover, the vertex set V is the only maximal -independent but is not a ( 1)-independent set. Thus every graph G satisfies ↵(G) = ↵0(G)  ↵1(G)  ↵2(G)  · · ·  ↵1(G) < ↵(G) = n. For k-independent set and k-independence number, Chellali, Favaron, Hansberg, and Volk- mann published a survey paper on this subject; see [3]. We must mention that the k- independence number of G is defined as the size of a largest k-colorable subgraph of G in [17]. In graph theory, Cartesian product, strong product, lexicographical product, and direct product are four of main products, each with its own set of applications and theoretical interpretations. Product networks were proposed based upon the idea of using the cross product as a tool for “combining” two known graphs with established properties to obtain a new one that inherits properties from both [5]. For more details on graph products, we refer to the book [10]. • The Cartesian product of two graphs G and H , written as G⇤H , is the graph with vertex set V (G)⇥V (H), in which two vertices (u, v) and (u0, v0) are adjacent if and only if u = u0 and (v, v0) 2 E(H), or v = v0 and (u, u0) 2 E(G). • The lexicographic product G H of graphs G and H has the vertex set V (G H) = V (G)⇥V (H). Two vertices (u, v), (u0, v0) are adjacent if uu0 2 E(G), or if u = u0 and vv0 2 E(H). • The strong product G ⇥ H of graphs G and H has the vertex set V (G) ⇥ V (H). Two vertices (u, v) and (u0, v0) are adjacent whenever uu0 2 E(G) and v = v0, or u = u0 and vv0 2 E(H), or uu0 2 E(G) and vv0 2 E(H). • The direct product G ⇥ H of graphs G and H has the vertex set V (G) ⇥ V (H). Two vertices (u, v) and (u0, v0) are adjacent if the projections on both coordinates are adjacent, i.e., uu0 2 E(G) and vv0 2 E(H). Note that unlike the other three products, the lexicographic product is a non-commuta- tive product since G H is usually not isomorphic to H G. For the independence number of Cartesian product graphs, Vizing [16] observed: E. Cheng et al.: The k-independence number of graph products 3 Theorem 1.1 ([10, 16]). For any graphs G and H , (i) ↵(G⇤H)  min{↵(G)|V (H)|,↵(H)|V (G)|}; (ii) ↵(G⇤H) ↵(G)↵(H) + min{|V (G)| ↵(G), |V (H)| ↵(H)}. Geller and Stahl [9] obtained the following result for the independence number of lexi- cographical product graphs. Theorem 1.2 ([9]). For any graphs G and H , ↵(G H) = ↵(G)↵(H). The following result is immediate, since G⇥H is a subgraph of G H . Corollary 1.3 ([10]). For any graphs G and H , ↵(G⇥H) ↵(G)↵(H). In 2011, S̆pacapan [17] proved the following theorem. Theorem 1.4 ([17]). For any graph G and H , (i) ↵(G⇥H) max{↵(G)|V (H)|,↵(H)|V (G)|}; (ii) ↵(G⇥H)  ↵(H)|V (G)|+ ↵(G)|V (H)| ↵(H)↵(G). For the independence number of four graph products, Jha and Slutzki obtained the following relation in 1994. Theorem 1.5 ([12]). For any graphs G and H , ↵(G H)  ↵(G⇥H)  ↵(G⇤H)  ↵(G⇥H). In this paper, we consider four standard products: the lexicographic, the strong, the Cartesian and the direct with respect to the k-independence number. Every of these four products will be treated in one of the forthcoming subsections in Section 2. Our results can be seen as extensions of Theorems 1.1, 1.2, 1.4, 1.5 and Corollary 1.3. 2 Main results In this section, let G and H be two connected graphs with V (G) = {u1, u2, . . . , un} and V (H) = {v1, v2, . . . , vm}, respectively. Then V (G⇤H) = {(ui, vj) | 1  i  n, 1  j  m}, where ⇤ denotes lexicographic product operation, strong product operation, Cartesian product operation or direct product operation. For v 2 V (H), we use G(v) to denote the subgraph of G ⇤ H induced by the vertex set {(ui, v) | 1  i  n}. Similarly, for u 2 V (G), we use H(u) to denote the subgraph of G ⇤ H induced by the vertex set {(u, vj) | 1  j  m}. 2.1 The lexicographic product In this subsection, we give upper and lower bounds of ↵k(G H). Theorem 2.1. (i) Let k 0 be an integer. For graphs G and H , ↵k(G H)  ↵k(H)|V (G)|. 4 Art Discrete Appl. Math. 1 (2018) #P1.01 (ii) Let k, r 0 be two integers. Let H be a graph of order m. For graphs G and H , ↵k(G H) ↵r(G)↵krm(H) where ↵krm(H) = 0 if k  rm. Moreover, the bounds are sharp. Proof. (i) Let I be a maximum k-independent set of GH . We claim that |I\V (H(ui))|  ↵k(H(ui)) for each ui 2 V (G). To see this, we observe that H(ui)[I \ V (H(ui))] is a subgraph of G H[I]. (ii) Let I be a maximum r-independent set of G, and J be a maximum (k rm)-inde- pendent set of H . Set I = {ui | 1  i  s} and J = {vj | 1  j  t}. For any (ui, vj) 2 I ⇥ J , we show that the degree of (ui, vj) in G H[I ⇥ J ] is at most k. Since I is a maximum r-independent set of G, it follows that dG[I](ui)  r, where ui 2 V (G[I]). Similarly, since J is a maximum (kmr)-independent set of H , it follows that dH[J](vj)  k mr, where vj 2 V (H[J ]). Then dGH[I⇥J](ui, vj)  dH[J](vj) +mdG[I](ui)  k mr +mr = k, and hence I ⇥ J is a k-independent set of G H . So ↵k(G H) ↵r(G)↵krm(H). See Remarks 2.4 and 2.5 for the sharpness. 2.2 The strong product In this subsection, we derive upper and lower bounds of ↵k(G⇥H). Theorem 2.2. (i) Let k 0 be an integer. For graphs G and H , ↵k(G⇥H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}. (ii) Let k, r 0 be two integers. For graphs G and H , ↵k(G⇥H) ↵r(G)↵b k2r+1 c(H) and ↵k(G⇥H) ↵r(H)↵b k2r+1 c(G). Moreover, the bounds are sharp. Proof. (i) Let I be a maximum k-independent set of G ⇥ H . If |G(vj) \ I| > ↵k(G) for some j  m, then I is not a k-independent set in G ⇥ H . It follows ↵k(G ⇥ H)  ↵k(G)|V (H)|. From the symmetry, we have ↵k(G⇥H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}. (ii) Let I be a maximum r-independent set of G, and J be a maximum ( k2r+1 )-inde- pendent set of H . Set I = {ui | 1  i  s} and J = {vj | 1  j  t}. E. Cheng et al.: The k-independence number of graph products 5 For any (ui, vj) 2 I ⇥ J , we show that the degree of (ui, vj) in G H[I ⇥ J ] is at most k. Since I is a maximum r-independent set of G, it follows that dG[I](ui)  r, where ui 2 V (G[I]). Similarly, since J is a maximum ( k2r+1 )-independent set of H , it follows that dH[J](vj)  k2r+1 , where vj 2 V (H[J ]). Then dG⇥H[I⇥J](ui, vj)  dH[J](vj) + 2k 2r + 1 dG[I](ui)  k 2r + 1 + 2rk 2r + 1 = k, and hence I ⇥ J is a k-independent set of G⇥H . So ↵k(G⇥H) ↵r(G)↵b k2r+1c(H). See Remarks 2.4 and 2.5 for the sharpness. 2.3 The Cartesian product Upper and lower bounds of ↵k(G⇤H) are derived in this subsection. Theorem 2.3. Let k, r 0 be two integers. For graphs G and H , (i) ↵k(G⇤H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}; (ii) ↵k(G⇤H) ↵r(G)↵kr(H) + 8 >>>>>>>>< >>>>>>>>: st, if k s+ t 2; t(k t+ 2), if s bk+32 c, t < b k+3 2 c, and k  s+ t 3; s(k s+ 2), if t bk+32 c, s < b k+3 2 c, and k  s+ t 3; min{p, q} ⌃ k 2 ⌥ + 1 ⌅ k 2 ⇧ + 1 , if s bk+32 c and t b k+3 2 c, where 0  r  k, s = |V (G)|↵r(G), t = |V (H)|↵kr(H), s = ( ⌃ k 2 ⌥ +1)p+s0, t = ( ⌅ k 2 ⇧ + 1)q + t0, 0  s0 < ⌃ k 2 ⌥ + 1 and 0  t0 < ⌅ k 2 ⇧ + 1. Moreover, the bounds are sharp. Proof. (i) The proof is similar to the proof of (i) of Theorem 2.1. (ii) Suppose I is a r-independent set in G and J is a (k r)-independent set in H , respectively. We will prove that I ⇥ J is a k-independent set of G⇤H . By commutativity, we may assume |V (G)|↵r(G)  |V (H)|↵kr(H). Say V (H)\J = {y1, y2, · · · , yt}, and take a subset {x1, x2, · · · , xs} ✓ V (G) \ I . Then s  t. Set K = {(xi, yj) | 1  i  s, 1  j  t}. Let F = G⇤H . Since F [K] is a spanning subgraph of Ks ⇤Kt, it follows that ↵k(F [K]) ↵k(Ks ⇤Kt), and hence there is a ↵k(Ks ⇤Kt)-independent set of F [K], say K 0. Claim 1: (I ⇥ J) [K 0 is a k-independent set of G⇤H . Proof of Claim 1. For any (ui, vj) 2 I ⇥ J where ui 2 V (G) and vj 2 V (H), we have dG⇤H[I⇥J](ui, vj) = dG[I](ui) + dH[J](vj)  r + (k r) = k. Therefore, I ⇥ J is a k-independent set of G⇤H . From the structure of Cartesian product graphs, we have EG⇤H [I ⇥ J,K 0] = ;. Then (I ⇥ J) [ K 0 is a k-independent set of G⇤H . 6 Art Discrete Appl. Math. 1 (2018) #P1.01 From Claim 1, we have ↵k(G⇤H) |(I⇥J)[K 0| = ↵r(G)↵kr(H)+↵k(Ks ⇤Kt) for graphs G and H . If k s+ t2, then (V (G) I)⇥ (V (H)J) = Ks ⇥ Kt is a k-independent set of Ks ⇤Kt, and hence ↵k(Ks ⇤Kt) st. If s bk+32 c, t < bk+32 c, and k  s+ t3, then ↵k(Ks ⇤Kt) ↵k(Kkt+2 ⇤Kt) t(k t + 2). Similarly, if t bk+32 c, s < bk+32 c, and k  s+ t3, then ↵k(Ks ⇤Kt) ↵k(Ks ⇤Kks+2) s(ks+2). If s bk+32 c, t bk+32 c, then ↵k(Ks ⇤Kt) ↵d k2 e(Ks)↵b k2 c(Kt) + ↵k(Ksd k2 e1 ⇤Ktb k2 c1) ✓⇠ k 2 ⇡ + 1 ◆✓ k 2 ⌫ + 1 ◆ + ↵k(Ksd k2 e1 ⇤Ktb k2 c1) ✓⇠ k 2 ⇡ + 1 ◆✓ k 2 ⌫ + 1 ◆ + ↵d k2 e(Ksd k2 e1)↵b k2 c(Ktb k2 c1) + ↵k(Ks2d k2 e2 ⇤Kt2b k2 c2) = 2 ✓⇠ k 2 ⇡ + 1 ◆✓ k 2 ⌫ + 1 ◆ + ↵k(Ks2d k2 e2 ⇤Kt2b k2 c2) = · · · = min{p, q} ✓⇠ k 2 ⇡ + 1 ◆✓ k 2 ⌫ + 1 ◆ + ↵k(Ksmin{p,q}(d k2 e+1) ⇤Ktmin{p,q}(b k2 c+1)) min{p, q} ✓⇠ k 2 ⇡ + 1 ◆✓ k 2 ⌫ + 1 ◆ , where s = ( ⌃ k 2 ⌥ +1)p+ s0, t = ( ⌅ k 2 ⇧ +1)q+ t0, 0  s0 < ⌃ k 2 ⌥ +1 and 0  t0 < ⌅ k 2 ⇧ +1. So the result follows. See Remarks 2.4 and 2.5 for the sharpness. Remark 2.4. From Theorems 2.1, 2.2 and 2.3, we have the following upper bounds for k-independent number. • ↵k(G H)  ↵k(H)|V (G)|; • ↵k(G⇥H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}; • ↵k(G⇤H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}. To show the sharpness of these upper bounds, we consider the following example. Let G = nK1 and |V (H)| = m. Then G ⇤ H consists of n copies of H , where ⇤ denotes the lexicographical or Cartesian or strong product operation. It is clear that ↵k(G ⇤H) = ↵k(H)n = ↵k(H)|V (G)|. So all these upper bounds are sharp. Remark 2.5. From Theorems 2.1, 2.2 and 2.3, we have the following lower bounds for k-independent number. • ↵k(G H) ↵r(G)↵krm(H), where m = |V (H)|; • ↵k(G⇥H) ↵r(G)↵b k2r+1 c(H); E. Cheng et al.: The k-independence number of graph products 7 • ↵k(G⇤H) ↵r(G)↵kr(H) + X , where s = |V (G)| ↵r(G), t = |V (H)| ↵kr(H), and X = 8 >>>>>>>>< >>>>>>>>: st, if k s+ t 2; t(k t+ 1), if s bk+32 c, t < b k+3 2 c, and k  s+ t 3; s(k s+ 1), if t bk+32 c, s < b k+3 2 c, and k  s+ t 3; min{p, q} ⌃ k 2 ⌥ + 1 ⌅ k 2 ⇧ + 1 , if s bk+32 c and t b k+3 2 c. To show the sharpness of these lower bounds, we first consider the following example. Let G = K2 and H = K2. Then G H = G ⇥H = K4, and ↵k(G H) = ↵k(G ⇥H) = ↵k(K4). For k = 0, ↵k(G H) = ↵k(G⇥H) = ↵0(K4) = 1; for k = 1, ↵k(G H) = ↵k(G⇥H) = ↵1(K4) = 2. From Theorems 2.1 and 2.2, ↵k(G H) ↵r(G)↵krm(H) and ↵k(G⇥H) ↵r(G)↵b k2r+1 c(H). Set r = 0. Then ↵k(G H) ↵0(K2)↵k(K2) = ↵k(K2) and ↵k(G ⇥ H) ↵0(K2)↵k(K2) = ↵k(K2). For k = 0, ↵0(K2) = 1; for k = 1, ↵1(K2) = 2. For k = 0, ↵k(G H) = ↵k(G⇥H) = ↵0(G)↵k(H). This implies that the first two lower bounds are sharp. Next, we consider the examples for Cartesian product. Let G = K2 and H = K2. Clearly, G⇤H = C4, and ↵k(G⇤H) = ↵k(C4). If k = 0, then r = 0, s = t = p = q = 1, and ↵k(G⇤H) = ↵0(C4) = 2 = ↵0(K2)↵0(K2) + st = ↵0(K2)↵0(K2) + min{p, q} ⌃ 0 2 ⌥ + 1 ⌅ 0 2 ⇧ + 1 . So the bound for the case k s+t2 or s bk+32 c, t bk+32 c is sharp. For the case s b k+3 2 c, t < b k+3 2 c, and k  s + t 3, we let G = K7 and H = K4. If k = 3, r = 2, s = 4, and t = 2, then ↵3(G⇤H) ↵2(K7)↵1(K4) + t(k t + 2) = 12. It suffices to show that ↵3(G⇤H)  12. Assume, to the contrary, that ↵3(G⇤H) 13. Let V (G) = V (K7) = {ui | 1  i  7} and V (H) = V (K4) = {vi | 1  i  4}. Then S4 i=1 V (G(vi)) = V (G⇤H). Let I be a maximum 3-independent set in G⇤H . Then |I| 13. Since k = 3, it follows that |I \ V (G(vi))|  4 for each i (1  i  4). Then there exists some G(vi) such that |I \ V (G(vi))| = 4. Without loss of generality, let I \ V (G(v1)) = {(uj , v1) | 1  j  4}. Since k = 3 and |I| 13, it follows that |I \ V (G(vi))| = 3 for each i (2  i  4). Since k = 3, it follows that I \ V (G(vi)) = {(uj , vi) | 5  j  7} for each i (2  i  4). Then the degree of the subgraph induced by I is at least 4, a contradiction. So ↵3(G⇤H) = 12, and hence the lower bound is also sharp. 2.4 The direct product We give upper and lower bounds for ↵k(G⇥H) in this section. Theorem 2.6. Let k 0 be an integers. For graphs G and H , (i) ↵k(G⇥H) max n ↵b k(H) c (G)|V (H)|,↵b k(G) c(H)|V (G)| o ; (ii) ↵k(G⇥H)  min n ↵b k(G) c (H)|V (G)|+ ↵b k (H) c (G)|V (H)| ↵b k(G) c(H)↵b k(H) c(G), ↵b k(G) c (G)|V (H)|+ ↵b k (H) c (H)|V (G)| ↵b k(H) c(G)↵b k(G) c(H) o . 8 Art Discrete Appl. Math. 1 (2018) #P1.01 Moreover, the bounds are sharp. Proof. (i) If I is a b k(H)c-independent set of G, then I ⇥ V (H) is a k-independent set of G ⇥ H . Therefore, ↵k(G ⇥ H) ↵b k(H) c(G)|V (H)|. By symmetry of direct product graphs, we have ↵k(G⇥H) max n ↵b k(H) c (G)|V (H)|,↵b k(G) c(H)|V (G)| o . (ii) Let I be a k-independent set of G ⇥ H . Partition I into two vertex subsets J,K such that J = ⇢ (u, v) 2 I | (u, vj) 2 I, vj 2 S(u,v), and |S(u,v)|  k (G) ⌫ and K = I \ J , where S(u,v) = {vj 2 NH(v) | (u, vj) 2 I}. Set J ui = J \H(ui) and Kvj = K \G(vj). Let IH be a maximum b k(G)c-independent set of H . Set Y = (V (G)⇥ IH) \K and Y ui = Y \H(ui) Note that Jui \ Y ui = ?. From the definition of J , Jui [ Y ui is a b k(G)c-independent set of H , and hence ↵b k(G) c (H) |Jui |+ |Y ui |. (2.1) Claim 1: For vj 2 V (H), ↵b k (H) c (G) |Kvj |. Proof of Claim 1. For (u, vj) 2 Kvj where u 2 V (G), from the definition of Kvj , we have dH(vj) > b k(G)c. Since dG(u) · dH(vj)  k, it follows that dG(u)  k dH(vj)  k (H) . Note that Kvj is a b k (H)c-independent set of G(vj). Therefore, ↵ k (H) (G) |Kvj |. Since P ui2V (G) |Y ui | = P vj2I(H) |K vj |, it follows from (2.1) and Claim 1 that X ui2V (G) (↵b k(G) c(H) |J ui |) + X vj2V (H) (↵b k (H) c (G) |Kvj |) X ui2V (G) |Y ui |+ X vj2V (H) (↵b k (H) c (G) |Kvj |) X vj2I(H) |Kvj |+ X vj2I(H) (↵b k (H) c (G) |Kvj |) X vj2I(H) ↵b k (H) c (G) E. Cheng et al.: The k-independence number of graph products 9 and hence X ui2V (G) ↵b k(G) c (H) + X vj2V (H) ↵b k (H) c (G) X vj2I(H) ↵b k (H) c (G) X ui2V (G) |Jui |+ X vj2V (H) |Kvj |. Then ↵b k(G) c (H)|V (G)|+ ↵b k (H) c (G)|V (H)| ↵b k(G) c(H)↵b k(H) c(G) |I|. From the symmetry of direct product, we have ↵k(G⇥H)  min n ↵b k(G) c (H)|V (G)|+ ↵b k (H) c (G)|V (H)| ↵b k(G) c(H)↵b k(H) c(G), ↵b k(G) c (G)|V (H)|+ ↵b k (H) c (H)|V (G)| ↵b k(H) c(G)↵b k(G) c(H) o . The proof is now complete. See Remark 2.7 for the sharpness. Remark 2.7. To show the sharpness of the lower and upper bounds in Theorem 2.6, we let G = K2 and H = K2. Then • ↵k(G⇥H) max{↵k(K2)|V (K2)|,↵k(K2)|V (K2)|} = 2↵k(K2); • ↵k(G⇥H)  min{↵k(H)|V (G)|+ ↵k(G)|V (H)| ↵k(H)↵k(G), ↵k(G)|V (H)|+ ↵k(H)|V (G)| ↵k(G)↵k(H)} = ↵k(K2)|V (K2)|+ ↵k(K2)|V (K2)| ↵k(K2)↵k(K2) = (4 ↵k(K2))↵k(K2). For k 1, we have ↵k(G ⇥ H) = 2, which implies that the upper and lower bounds in Theorem 2.6 are sharp. 2.5 Relation of four graph products For the k-independence number of four graph products, we have the following relation. Proposition 2.8. For any graphs G and H , ↵k(G H)  ↵k(G⇥H)  ↵k(G⇤H)  min{↵k(H)(G⇥H),↵k(G)(G⇥H)}. Proof. Since G ⇥ H is a subgraph of G H , it follows that ↵k(G H)  ↵k(G ⇥ H). Similarly, since G⇤H is a subgraph of G⇥H , it follows that ↵k(G⇥H)  ↵k(G⇤H). From Theorem 2.3, ↵k(G⇤H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}. From Theorem 2.6, we have ↵k(H)(G⇥H) max{↵k(G)|V (H)|,↵ k(H) (G) (H)|V (G)|} min{↵k(G)|V (H)|,↵k(H)|V (G)|} ↵k(G⇤H). Similarly, we have ↵k(G)(G⇥H) ↵k(G⇤H), and hence ↵k(G⇤H)  min{↵k(H)(G⇥H),↵k(G)(G⇥H)}. The proof is now complete. 10 Art Discrete Appl. Math. 1 (2018) #P1.01 3 Applications In this section, we demonstrate the usefulness of the proposed constructions by applying them to some instances of Cartesian and lexicographical product networks. The following results are immediate. Proposition 3.1. Let k 0, n 2 be two integers and {n3 } be the integer such that n ⌘ {n3 }(mod 3). (i) For a complete graph Kn, ↵k(Kn) = ( k + 1, if 0  k  n 1; n, if k n. (ii) For a path Pn, ↵k(Pn) = 8 >< >: dn2 e, if k = 0; 2bn3 c+ { n 3 }, if k = 1; n, if k 2. (iii) For a cycle Cn, ↵k(Cn) = 8 >>>< >>>: bn2 c, if k = 0; 2bn3 c, if k = 1 and n ⌘ 0, 1 (mod 3); 2bn3 c+ 1, if k = 1 and n ⌘ 2 (mod 3); n, if k 2. 3.1 n-dimensional generalized hypercube Let Km be a clique of m vertices, m 2. An n-dimensional generalized hypercube [5, 8] is the product of n cliques. We first focus our attention on 2-dimensional generalized hypercube. Proposition 3.2. For network Km1⇤Km2 , min{m1, dk/2e+ 1}min{m2, bk/2c+ 1}  ↵k(Km1⇤Km2)  8 >>>< >>>: min{m2,m1}(k + 1), if k  mi 1 (i = 1, 2); (k + 1)m1, if k  m2 1, k m1; (k + 1)m2, if k  m1 1, k m2; m1m2, if k m1, k m2. Proof. We first investigate the upper bound of ↵k(Km1⇤Km2). If k mi (i = 1, 2), then ↵k(Kmi) = mi and ↵k(Km1⇤Km2)  min{↵k(Km1)|V (Km2)|,↵k(Km2)|V (Km1)|} = m1m2 by Theorem 2.3. If k  m2 1 and k m1, then ↵k(Km1) = m1 and ↵k(Km2)=k+1 and ↵k(Km1⇤Km2)  min{↵k(Km1)|V (Km2)|,↵k(Km2)|V (Km1)|} = min{m1m2, (k + 1)m1} = (k + 1)m1. Similarly, if k  m1 1 and k m2, then ↵k(Km1⇤Km2)  (k + 1)m2. If k  mi 1 (i = 1, 2), then ↵k(Kmi) = k + 1, and hence ↵k(Km1⇤Km2)  min{(k + 1)m2, (k + 1)m1} = min{m2,m1}(k + 1). E. Cheng et al.: The k-independence number of graph products 11 Next, we consider the lower bound of ↵k(Km1⇤Km2). From Theorem 2.3, we have ↵k(Km1⇤Km2) ↵r(Km1)↵kr(Km2), where 0  r  k. If r = dk/2e, then k r = bk/2c, ↵r(Km1) = min{m1, dk/2e + 1}, and ↵kr(Km2) = min{m2, bk/2c + 1}. Furthermore, we have ↵k(Km1⇤Km2) ↵r(Km1)↵kr(Km2) = min{m1, dk/2e + 1}min{m2, bk/2c+ 1}, as desired. Next, we consider n-dimensional generalized hypercube. Proposition 3.3. For network Km1⇤Km2⇤ · · ·⇤Kmn , we have the following. (i) If mi  k (1  i  n), then m1  ↵k(Km1⇤Km2⇤ · · ·⇤Kmn)  nY i=1 mi. (ii) If k  mj 1 (1  j  n), then k + 1  ↵k(Km1⇤Km2⇤ · · ·⇤Kmn)  (k + 1) nY i=2 mi. Proof. (i) Since mi  k (1  i  n), it follows that ↵k(Kmi) = mi, where 1  i  n. From Theorem 2.3, we have ↵k(G⇤H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}  ↵k(G)|V (H)| for any two graphs G and H , and hence ↵k(Km1⇤Km2⇤ · · ·⇤Kmn) = ↵k((Km1⇤Km2⇤ · · ·⇤Kmn1)⇤Kmn)  ↵k((Km1⇤Km2⇤ · · ·⇤Kmn1)mn  ↵k((Km1⇤Km2⇤ · · ·⇤Kmn2)mn1mn  . . .  ↵k(Km1)m2 · · ·mn1mn = nY i=1 mi. From Theorem 2.3, we have ↵k(G⇤H) ↵r(G)↵kr(H) for any two graphs G and H . Set r = k. Then ↵k(G⇤H) ↵k(G)↵0(H) for any two graphs G and H , and hence ↵k(Km1⇤Km2⇤ · · ·⇤Kmn) = ↵k((Km1⇤Km2⇤ · · ·⇤Kmn1)⇤Kmn) ↵k(Km1⇤Km2⇤ · · ·⇤Kmn1)↵0(Kmn) ↵k(Km1⇤Km2⇤ · · ·⇤Kmn1) ↵k((Km1⇤Km2⇤ · · ·⇤Kmn2) . . . ↵k(Km1) = m1. (ii) Since k  mj 1 (1  j  n), it follows that ↵k(Kmj ) = k + 1, where 1  j  n. From Theorem 2.3, we have ↵k(G⇤H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}  12 Art Discrete Appl. Math. 1 (2018) #P1.01 ↵k(G)|V (H)| for any two graphs G and H , and hence ↵k(Km1⇤Km2⇤ · · ·⇤Kmn) = ↵k((Km1⇤Km2⇤ · · ·⇤Kmn1)⇤Kmn)  ↵k((Km1⇤Km2⇤ · · ·⇤Kmn1)mn  ↵k((Km1⇤Km2⇤ · · ·⇤Kmn2)mn1mn  . . .  ↵k((Km1)m2 · · ·mn1mn = (k + 1) nY i=2 mi. From Theorem 2.3, we have ↵k(G⇤H) ↵r(G)↵kr(H) for any two graphs G and H . Set r = k. Then ↵k(G⇤H) ↵k(G)↵0(H) for any two graphs G and H , and hence ↵k(Km1⇤Km2⇤ · · ·⇤Kmn) = ↵k((Km1⇤Km2⇤ · · ·⇤Kmn1)⇤Kmn) ↵k(Km1⇤Km2⇤ · · ·⇤Kmn1)↵0(Kmn) ↵k(Km1⇤Km2⇤ · · ·⇤Kmn1) ↵k((Km1⇤Km2⇤ · · ·⇤Kmn2) . . . ↵k(Km1) = k + 1, as desired. Proposition 3.4. For network Km1 Km2 · · · Kmn , ↵k(Km1 Km2 · · · Kmn) = ( k + 1, if 0  k  Q n i=1 mi 1;Q n i=1 mi, if k P n i=1 mi. Proof. From the definition of lexicographical product, Km1Km2· · ·Kmn is a complete graph. From Proposition 3.1, if 0  k  Q n i=1 mi1, then ↵k(Km1 Km2 · · ·Kmn) = k + 1; if k + 1k P n i=1 mi, then ↵k(Km1 Km2 · · · Kmn) = Q n i=1 mi. 3.2 Two-dimensional grid graph A two-dimensional grid graph is the Cartesian product Pn ⇤Pm of path graphs on m and n vertices. For more details on grid graph, we refer to [2, 11]. The network Pn Pm is the lexicographical product Pn Pm of path graphs on m and n vertices; see [15]. Let {m/3} be the integer such that m ⌘ {m/3}(mod 3). Proposition 3.5. For network Pn ⇤Pm (n 3,m 3), we have the following. (i) If k 4, then ↵k(Pn ⇤Pm) = mn. (ii) If k = 2, 3, then min{mdn/2e, ndm/2e}  ↵k(Pn ⇤Pm)  mn. (iii) If k = 1, then dn/2e(2bm/3c+ {m/3})  ↵k(Pn ⇤Pm)  min{(2bn/3c+ {n/3})m, (2bm/3c+ {m/3})n}. E. Cheng et al.: The k-independence number of graph products 13 (iv) If k = 0, then dn/2edm/2e  ↵k(Pn⇤Pm)  min{dn/2em, dm/2en}. Proof. (i) Choose all vertices in Pn ⇤Pm. Since the degree of each vertex in the induced subgraph induced by these vertices is at most 4, it follows that ↵k(Pn ⇤Pm) = mn. (ii) From Theorem 2.3, ↵2(Pn ⇤Pm)  min{↵2(Pn)|V (Pm)|, ↵2(Pm)|V (Pn)|} = min{nm,mn} = mn and ↵2(Pn ⇤Pm) ↵r(Pn)↵2r(Pm). If r = 0, then we have ↵2(Pn ⇤Pm) ↵0(Pn)↵2(Pm) = dn/2em. If r = 2, then ↵2(Pn ⇤Pm) ↵2(Pn)↵0(Pm) = dm/2en. So, we have ↵2(Pn ⇤Pm) min{mdn/2e, ndm/2e}. Sim- ilarly, if k = 3, then min{mdn/2e, ndm/2e}  ↵3(Pn ⇤Pm)  mn. (iii) From Theorem 2.3, ↵1(Pn ⇤Pm)  min{↵1(Pn)|V (Pm)|,↵1(Pm)|V (Pn)|} = min{(2bn/3c + {n/3})m, (2bm/3c + {m/3})n}. From Theorem 2.3, ↵1(Pn ⇤Pm) ↵r(Pn)↵1r(Pm). If r = 0, then ↵1(Pn ⇤Pm) ↵0(Pn)↵1(Pm) = dn/2e(2bm/3c + {m/3}). (iv) From Theorem 2.3, ↵0(Pn ⇤Pm)  min{↵0(Pn)|V (Pm)|,↵0(Pm)|V (Pn)|} = min{dn/2em, dm/2en}, and ↵0(Pn ⇤Pm) ↵0(Pn)↵0(Pm) = dn/2edm/2e. Proposition 3.6. For network Pn Pm (n 4,m 3), we have the following. (i) If k 2m+ 2, then ↵k(Pn Pm) = mn. (ii) If 2  k < 2m+ 2, then dn/2em  ↵k(Pn Pm)  mn. (iii) If k = 1, then dn/2e(2bm/3c+ {m/3})  ↵1(Pn Pm)  n(2bm/3c+ {m/3}). (iv) If k = 0, then dn/2edm/2e  ↵k(Pn Pm)  ndm/2e. Proof. From Theorem 2.1, we have ↵k(Pn Pm)  ↵k(Pm)|V (Pn)| = n↵k(Pm) and ↵k(Pn Pm) ↵r(Pn)↵krm(Pm). Let r = 0. Then ↵k(Pn Pm) ↵0(Pn)↵k(Pm), and hence dn/2e↵k(Pm)  ↵k(Pn Pm)  n↵k(Pm). (3.1) (i) For k 2m+ 2, we choose all vertices in Pn Pm. Since the degree of each vertex in the induced subgraph induced by these vertices is at most 2m+ 2, it follows that ↵k(Pn Pm) = mn. (ii) Since 2  k < 2m + 2, it follows that ↵k(Pm) = m. From (3.1), dn/2em  ↵k(Pn Pm)  mn. (iii) For k = 1, ↵k(Pm) = 2bm/3c+{m/3}. From (3.1), dn/2e(2bm/3c+{m/3})  ↵1(Pn Pm)  n(2bm/3c+ {m/3}). (iv) For k = 0, ↵k(Pm) = bm/2c. From (3.1), we have dn/2edm/2e  ↵k(Pn Pm)  ndm/2e. 3.3 n-dimensional mesh An n-dimensional mesh is the Cartesian product of n paths. By this definition, two- dimensional grid graph is a 2-dimensional mesh. An n-dimensional hypercube is a special case of an n-dimensional mesh, in which the n linear arrays are all of size 2; see [13]. 14 Art Discrete Appl. Math. 1 (2018) #P1.01 Proposition 3.7. For n-dimensional mesh Pm1⇤Pm2⇤ · · ·⇤Pmn , ↵k(Pm1⇤Pm2⇤ · · ·⇤Pmn)  8 >< >: dm12 e Q n i=2 mi, if k = 0; (2bm13 c+ { m1 3 }) Q n i=2 mi, if k = 1;Q n i=1 mi, if k 2, and ↵k(Pm1⇤Pm2⇤ · · ·⇤Pmn) 8 >< >: dm12 e Q n i=2dmi/2e, if k = 0; (2bm13 c+ { m1 3 }) Q n i=2dmi/2e, if k = 1; m1 Q n i=2dmi/2e, if k 2. Proof. From Theorem 2.3, we have ↵k(G⇤H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}  ↵k(G)|V (H)| for any two graphs G and H , and hence ↵k(Pm1⇤Pm2⇤ · · ·⇤Pmn) = ↵k((Pm1⇤Pm2⇤ · · ·⇤Pmn1)⇤Pmn)  ↵k((Pm1⇤Pm2⇤ · · ·⇤Pmn1)mn  ↵k((Pm1⇤Pm2⇤ · · ·⇤Pmn2)mn1mn  . . .  ↵k(Pm1)m2 · · ·mn1mn. So, the result follows. From Theorem 2.3, we have ↵k(G⇤H) ↵r(G)↵kr(H) for any two graphs G and H . Set r = k. Then ↵k(G⇤H) ↵k(G)↵0(H) for any two graphs G and H , and hence ↵k(Pm1⇤Pm2⇤ · · ·⇤Pmn) = ↵k((Pm1⇤Pm2⇤ · · ·⇤Pmn1)⇤Pmn) ↵k(Pm1⇤Pm2⇤ · · ·⇤Pmn1)↵0(Pmn) ↵k(Pm1⇤Pm2⇤ · · ·⇤Pmn1)dmn/2e ↵k((Pm1⇤Pm2⇤ · · ·⇤Pmn2)dmn1/2edmn/2e . . . ↵k(Pm1) nY i=2 dmi/2e, and hence the result holds. Similarly to the proof of Proposition 3.7, we can obtain the following result. Proposition 3.8. For n-dimensional mesh Pm1 Pm2 · · · Pmn , 8 >< >: dm12 e  ↵k(Pm1 · · · Pmn)  d m1 2 e Q n i=2 mi, if k = 0; 2bm13 c+ { m1 3 }  ↵k(Pm1 · · · Pmn)  (2b m1 3 c+ { m1 3 }) Q n i=2 mi, if k = 1; m1  ↵k(Pm1 · · · Pmn)  Q n i=1 mi, if k 2. E. Cheng et al.: The k-independence number of graph products 15 3.4 n-dimensional torus An n-dimensional torus is the Cartesian product of n cycles Cm1 , Cm2 , · · · , Cmn of size at least three. The cycles Cmi are not necessary to have the same size. Ku et al. [14] showed that there are n edge-disjoint spanning trees in an n-dimensional torus. The network Cm1 Cm2 · · · Cmn is investigated in [15]. Here, we consider the networks constructed by Cm1⇤Cm2⇤ · · ·⇤Cmn and Cm1 Cm2 · · · Cmn , respectively. Proposition 3.9. For network Cn⇤Cm (n 3,m 3), we have the following. (i) If k 4, then ↵k(Cn⇤Cm) = mn. (ii) If k = 3 or k = 2, then min{mbn/2c, nbm/2c}  ↵k(Cn⇤Cm)  mn. (iii) If k = 1, then 2bn/2cbm3 c  ↵k(Cn⇤Cm)  min{m(2bn3 c+ 1), n(2bm3 c+ 1)}. (iv) If k = 0, then bn/2cbm/2c  ↵k(Cn⇤Cm)  min{bn/2cm, bm/2cn}. Proof. (i) Choose all vertices in Cn⇤Cm. Since the degree of each vertex in the induced subgraph induced by these vertices is at most 4, it follows that ↵k(Cn⇤Cm) = mn. (ii) From Theorem 2.3, ↵3(Cn⇤Cm)  min{↵3(Cn)|V (Cm)|,↵3(Cm)|V (Cn)|} = min{nm,mn} = mn, and ↵3(Pn⇤Pm) ↵r(Cn)↵3r(Cm). If r = 0, then we have ↵3(Cn⇤Cm) ↵0(Cn)↵3(Cm) = bn/2cm. If r = 3, then ↵3(Cn⇤Cm) ↵3(Cn)↵0(Cm) = bm/2cn = bm/2cn. So, ↵3(Cn⇤Cm) min{mbn/2c, nbm/2c}. The case k = 2 can be similarly proved. (iii) From Theorem 2.3, ↵1(Cn⇤Cm) ↵r(Cn)↵1r(Cm). If r = 0, then we have ↵1(Cn⇤Cm) ↵0(Cn)↵1(Cm) = bn/2c(2bm3 c), and ↵1(Cn⇤Cm)  min{↵1(Cn) |V (Cm)|,↵1(Cm)|V (Cn)|} = min{m(2bn3 c+ 1), n(2b m 3 c+ 1)}. (iv) From Theorem 2.3, ↵0(Cn⇤Cm)  min{bn/2cm, bm/2cn}, and ↵0(Cn⇤Cm) ↵0(Cn)↵0(Cm) = bn/2cbm/2c. For network Cn Cm, we have the following result. Proposition 3.10. For network Cn Cm (n 4,m 3), we have the following. (i) If k 2m+ 2, then ↵k(Cn Cm) = mn. (ii) If 2  k < 2m+ 2, then bn/2cm  ↵k(Cn Cm)  mn. (iii) If k = 1 and n ⌘ 0, 1 (mod 3), then 2bn/2cbn/3c  ↵k(Cn Cm)  2nbn/3c. (iv) If k = 1 and n ⌘ 2 (mod 3), then bn/2c(2bn/3c+ 1)  ↵k(Cn Cm)  n(2bn/3c+ 1). (v) If k = 0, then bm/2cbn/2c  ↵0(Cn Cm)  nbm/2c. Proof. From Theorem 2.1, we have ↵k(Cn Cm)  ↵k(Cm)|V (Cn)| = n↵k(Cm) and ↵k(Cn Cm) ↵r(Cn)↵krm(Cm). Let r = 0. Then ↵k(Cn Cm) ↵0(Cn)↵k(Cm), and hence bn/2c↵k(Cm)  ↵k(Cn Cm)  n↵k(Cm). (3.2) (i) For k 2m + 2, we choose all vertices in Cn Cm. Since the degree of each vertex in the induced subgraph induced by these vertices is at most 2m+ 2, it follows that ↵k(Cn Cm) = mn. 16 Art Discrete Appl. Math. 1 (2018) #P1.01 (ii) Since 2  k < 2m + 2, it follows that ↵k(Cm) = m, and hence bn/2cm  ↵k(Cn Cm)  mn by (3.2). (iii) Since k = 1 and n ⌘ 0, 1 (mod 3), we have ↵k(Cm) = 2bn3 c. From (3.2), 2bn/2cbn/3c  ↵k(Cn Cm)  2nbn/3c. (iv) For k = 1 and n ⌘ 2 (mod 3), ↵k(Cm) = 2bn3 c+1. From (3.2), bn/2c(2bn/3c+ 1)  ↵k(Cn Cm)  n(2bn/3c+ 1). (v) For k = 0, ↵k(Cm) = bm/2c. From (3.2), bm/2cbn/2c  ↵k(Cn Cm)  nbm/2c. For general case, we have the following two results. Proposition 3.11. For network Cm1⇤Cm2⇤ · · ·⇤Cmn , ↵k(Cm1⇤Cm2⇤ · · ·⇤Cmn) 8 >>>< >>>: bm12 c Q n i=2 mi, if k = 0; 2bm13 c Q n i=2 mi, if k = 1,m1 ⌘ 0, 1 (mod 3); (2bm13 c+ 1) Q n i=2 mi, if k = 1,m1 ⌘ 2 (mod 3);Q n i=1 mi, if k 2, and ↵k(Cm1⇤ · · ·⇤Cmn) 8 >>>< >>>: bm12 c Q n i=2bmi/2c, if k = 0; 2bm13 c Q n i=2bmi/2c, if k = 1,m1 ⌘ 0, 1 (mod 3); (2bm13 c+ 1) Q n i=2bmi/2c, if k = 1,m1 ⌘ 2 (mod 3); m1 Q n i=2bmi/2c, if k 2, where mi is the order of Cmi and 1  i  n. Proof. From Theorem 2.3, we have ↵k(G⇤H)  min{↵k(G)|V (H)|,↵k(H)|V (G)|}  ↵k(G)|V (H)| for any two graphs G and H , and hence ↵k(Cm1⇤Cm2⇤ · · ·⇤Cmn) = ↵k((Cm1⇤Cm2⇤ · · ·⇤Cmn1)⇤Cmn)  ↵k((Cm1⇤Cm2⇤ · · ·⇤Cmn1)mn  ↵k((Cm1⇤Cm2⇤ · · ·⇤Cmn2)mn1mn  . . .  ↵k(Cm1)m2 · · ·mn1mn. From (iii) of Proposition 3.1, the result follows. From Theorem 2.3, we have ↵k(G⇤H) ↵r(G)↵kr(H) for any two graphs G and H . Set r = k. Then ↵k(G⇤H) ↵k(G)↵0(H) for any two graphs G and H , and hence ↵k(Cm1⇤Cm2⇤ · · ·⇤Cmn) = ↵k((Cm1⇤Cm2⇤ · · ·⇤Cmn1)⇤Cmn) ↵k(Cm1⇤Cm2⇤ · · ·⇤Cmn1)↵0(Cmn) ↵k(Cm1⇤Cm2⇤ · · ·⇤Cmn1)bmn/2c ↵k((Cm1⇤Cm2⇤ · · ·⇤Cmn2)bmn1/2cbmn/2c . . . ↵k(Cm1) nY i=2 bmi/2c. From (3.2) of Proposition 3.1, the result holds. E. Cheng et al.: The k-independence number of graph products 17 Similarly to the proof of Proposition 3.11, we can prove the following result. Proposition 3.12. For network Cm1 Cm2 · · · Cmn , 8 >>>>< >>>>: bm12 c  ↵k(Cm1 · · · Cmn)  b m1 2 c Qn i=2 mi, if k = 0; 2bm13 c  ↵k(Cm1 · · · Cmn)  2b m1 3 c Qn i=2 mi, if k = 1 and m1 ⌘ 0, 1 (mod 3); 2bm13 c+ 1  ↵k(Cm1 · · · Cmn)  (2b m1 3 c+ 1) Qn i=2 mi, if k = 1 and m1 ⌘ 2 (mod 3); m1  ↵k(Cm1 · · · Cmn)  Qn i=1 mi, if k 2, where mi is the order of Cmi and 1  i  n. 3.5 n-dimensional hyper Petersen network An n-dimensional hyper Petersen network HPn is the product of the well-known Petersen graph and Qn3 [4], where n 3 and Qn3 denotes an (n 3)-dimensional hypercube. Note that HP3 is just the Petersen graph. The network HLn is the lexicographical product of the Petersen graph and Qn3, where n 3 and Qn3 denotes an (n3)-dimensional hypercube; see [15]. Note that HL3 is just the Petersen graph, and HL4 is a graph obtained from two copies of the Petersen graph by adding the edges between all the vertices from different copies of the Petersen graph. Proposition 3.13. (i) For network HP3 and HL3, ↵k(HP3) = ↵k(HL3) = 8 >>>< >>>: 4, if k = 0; 5, if k = 1; 5, if k = 2; 10, if k 3. (ii) For network HP4, 8 >>>>>>< >>>>>>: 5  ↵k(HP4)  8, if k = 0; 6  ↵k(HP4)  10, if k = 1; 6  ↵k(HP4)  10, if k = 2; 11  ↵k(HP4)  20, if k = 3; ↵k(HP4) = 30, if k 4. (iii) For network HL4, ( 4  ↵k(HP4)  15, if k = 0; 8  ↵k(HP4)  30, if k 1. Proof. (i) Note that HL3 or HP3 is just the Petersen graph, and its maximum degree is 3. Since |V (HP3)| = 10, it follows that ↵k(HP3) = 10 for k 3. One can also check that ↵k(HP3) = ↵k(HL3) = 8 >< >: 4, if k = 0; 5, if k = 1; 5, if k = 2. 18 Art Discrete Appl. Math. 1 (2018) #P1.01 (ii) For network HP4, HP4 = HP3⇤K2. From Theorem 2.3, we have ↵k(HP4) = ↵k(HP3⇤K2)  min{2↵k(HP3), 10↵k(K2)}. Note that ↵k(K2) = 1 for k = 0; ↵k(K2) = 2 for k 1. Combining this with (i) of this proposition, we have ↵k(HP4)  8 >>>< >>>: 8, if k = 0; 10, if k = 1; 10, if k = 2; 20, if k 3. From Theorem 2.3, ↵k(HP4) ↵r(HP3)↵kr(K2)+↵k(Ks⇤Kt), where s= |V (HP3)| ↵r(HP3) and t = |V (K2)| ↵kr(K2). Set r = k. Then t = 1 and ↵k(HP4) ↵k(HP3)↵0(K2) + ↵k(Ks⇤K1) ↵k(HP3) + 1, and hence ↵k(HP4) 8 >>>< >>>: 5, if k = 0; 6, if k = 1; 6, if k = 2; 11, if k 3. (iii) For network HL4, HL4 = K2 HL3. From Theorem 2.3, we have ↵k(HL4) = ↵k(K2 HL3)  |V (HL3)|↵k(K2) = 10↵k(K2). Note that ↵k(K2) = 1 for k = 0; ↵k(K2) = 2 for k 1. Combining this with (i) of this proposition, we have ↵k(HL4)  ( 15, if k = 0; 20, if k 1. From Theorem 2.3, ↵k(HL4) ↵r(HL3)↵k2r(K2). Set r = 0. Then ↵k(HL4) ↵0(HL3)↵k(K2) = 4↵k(K2), and hence ↵k(HL4) ( 4, if k = 0; 8, if k 1. References [1] J. A. Bondy and U. S. R. Murty, Graph theory, volume 244 of Graduate Texts in Mathematics, Springer, New York, 2008, doi:10.1007/978-1-84628-970-5. [2] N. J. Calkin and H. S. Wilf, The number of independent sets in a grid graph, SIAM J. Discrete Math. 11 (1998), 54–60, doi:10.1137/s089548019528993x. [3] M. Chellali, O. Favaron, A. Hansberg and L. Volkmann, k-domination and k-independence in graphs: a survey, Graphs Combin. 28 (2012), 1–55, doi:10.1007/s00373-011-1040-3. [4] S. K. Das, S. R. Öhring and A. K. Banerjee, Embeddings into hyper Petersen network: yet another hypercube-like interconnection topology, VLSI Design 2 (1995), 335–351, doi: 10.1155/1995/95759. [5] K. Day and A.-E. Al-Ayyoub, The cross product of interconnection networks, IEEE Trans. Parallel Distrib. Syst. 8 (1997), 109–118, doi:10.1109/71.577251. E. Cheng et al.: The k-independence number of graph products 19 [6] J. F. Fink and M. S. Jacobson, n-domination in graphs, in: Graph theory with applications to algorithms and computer science, Wiley, New York, Wiley-Intersci. Publ., pp. 283–300, 1985, http://dl.acm.org/citation.cfm?id=21936.25446. [7] J. F. Fink and M. S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, in: Graph theory with applications to algorithms and computer science, Wiley, New York, Wiley- Intersci. Publ., pp. 301–311, 1985, http://dl.acm.org/citation.cfm?id=25447. [8] P. Fragopoulou, S. Akl and H. Meijer, Optimal communication primitives on the generalized hypercube network, J. Parallel Distrib. Comput. 32 (1996), 173–187, doi:10.1006/jpdc.1996. 0012. [9] D. Geller and S. Stahl, The chromatic number and other functions of the lexicographic product, J. Combinatorial Theory Ser. B 19 (1975), 87–95, doi:10.1016/0095-8956(75)90076-3. [10] R. Hammack, W. Imrich and S. Klavžar, Handbook of product graphs, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2nd edition, 2011, with a foreword by Peter Winkler. [11] A. Itai and M. Rodeh, The multi-tree approach to reliability in distributed networks, Inform. and Comput. 79 (1988), 43–59, doi:10.1016/0890-5401(88)90016-8. [12] P. K. Jha and G. Slutzki, Independence numbers of product graphs, Appl. Math. Lett. 7 (1994), 91–94, doi:10.1016/0893-9659(94)90018-3. [13] S. L. Johnsson and C.-T. Ho, Optimum broadcasting and personalized communication in hy- percubes, IEEE Trans. Comput. 38 (1989), 1249–1268, doi:10.1109/12.29465. [14] S. Ku, B. Wang and T. Hung, Constructing edge-disjoint spanning trees in product networks, IEEE Trans. Parallel Distrib. Syst. 14 (2003), 213–221, doi:10.1109/tpds.2003.1189580. [15] Y. Mao, Path-connectivity of lexicographic product graphs, Int. J. Comput. Math. 93 (2016), 27–39, doi:10.1080/00207160.2014.987762. [16] V. G. Vizing, The cartesian product of graphs, Vyčisl. Sistemy No. 9 (1963), 30–43. [17] S. Špacapan, The k-independence number of direct products of graphs and Hedetniemi’s con- jecture, European J. Combin. 32 (2011), 1377–1383, doi:10.1016/j.ejc.2011.07.002. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.02 https://doi.org/10.26493/2590-9770.1186.258 (Also available at http://adam-journal.eu) Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization Jürgen Bokowski Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany Michael Cuntz Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany Dedicated to Prof. Dr. Dr. h.c. Jörg M. Wills on the Occation of his 80th Birthday. Received 19 July 2017, accepted 13 August 2017, published online 10 November 2017 Abstract A Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g 1) automorphisms, where g is the genus of the surface. The Hurwitz surface of least genus is the Klein quartic of genus 3. A polyhedral realization without self-intersections of Klein’s quartic of genus 3 was found by E. Schulte and J. M. Wills in 1985. For the next possible genus of a Hurwitz surface, i.e., for the genus 7 case with 72 vertices, we provide a polyhedral realization without self-intersections. We also show a topological representation for which we have a corresponding model. Keywords: Hurwitz surface, regular map, polyhedral manifold. Math. Subj. Class.: 52B70 E-mail addresses: juergen.bokowski@gmail.com (Jürgen Bokowski), cuntz@math.uni-hannover.de (Michael Cuntz) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.02 1 Introduction In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g 1) automorphisms, where g is the genus of the surface. The Hurwitz surface of least genus is the Klein quartic of genus 3. The next possible genus is 7 with automorphism group PSL(2, 8), which is the simple group of order 84⇥ (71) = 504; if one includes orientation-reversing isometries, the group is of order 1 008. Our paper is devoted to this genus 7 surface of Adolf Hurwitz from 1893, compare [12], which provides us in modern terminology with a regular map of type (3, 7)18. We have a closed triangular 2-manifold in which each vertex is incident with seven triangles. The Petrie polygon length is 18 and the automorphism group is flag transitive. In general a regular map is a decomposition of a two dimensional manifold into topological discs, such that every flag can be transformed into any other flag by a symmetry of the decomposition. When we describe a topological disc d via a circular sequence of its vertices d = (v1, v2, . . . , vk), a flag will be in this context a tripel (vi, (vi, vi+1), d) consisting of a vertex vi, an edge (vi, vi+1), and the disc d itself. For the Hurwitz surface of genus 7, the name Macbeath surface is used as well, although the corresponding article of Macbeath is from 1965, [13]. We find under Wikipedia for regular map: “regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph-theoretically”. How- ever, there are also results in which polyhedral realizations of regular maps have been studied, see e.g. the corresponding articles of Jörg M. Wills and of his co-authors or other colleagues in [2, 3, 4, 5, 6, 7, 8, 16, 17], and [18]. This article is devoted to such a question that was studied by Jörg M. Wills for some time. When only some abstract combinatorial data of a geometric object is given and when we are looking for a corresponding geometric realization or try to prove that no such realization exists, we are facing in general a hard problem that has been called a problem of computational synthetic geometry in [4]. Our main result of this article provides a polyhedral realization of Hurwitz’s regular map (3, 7)18 of genus seven. We also show a topological representation for which we have a corresponding 3D-model. We refer the reader for additional aspects with respect to this paper to the homepage of the second author: http://www.iazd.uni-hannover. de/cuntz.html. 1.1 Previous polyhedral realizations of regular maps Regular maps generalize on a combinatorial level Platonic solids with their geometric flag transitive automorphism groups. Mani’s result [14] asserts that for each combinatorial automorphism of the boundary structure of a convex polyhedron, there does exist a convex polyhedron with a corresponding geometric symmetry. A corresponding result for general regular maps does not hold, the notion hidden symmetries has been used. The polyhedral realization of Hurwitz’s surface of least genus, i.e., a polyhedral real- ization of Klein’s quartic of genus 3, with 24 vertices has been published by E. Schulte and J. M. Wills in [16]. In Figure 1 we have depicted two truncated tetrahedra the vertices of which are the vertices of this symmetric realization. A first polyhedral realization of a regular map of Walther Dyck (3, 8)6 with 12 vertices was provided in Antibes in 1987 by Bokowski, see [1] and [2], thus disproving a conjecture of Schulte and Wills that it did not exist. A symmetrical version of this map was found later by Ulrich Brehm, [6]. U. Brehm and U. Leopold have found another polyhedral realization J. Bokowski and M. Cuntz: Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization 3 Figure 1: The points of a realization of Hurwitz’s regular map of genus 3 by E. Schulte and J. M. Wills are the vertices of two truncated tetrahedra, [16]. Figure 2: First realization of Dyck’s regular map of genus 3 presented by Bokowski (in the middle) in Antibes 1987, [1]. The other two men in the photo are R. Connelly, Cornell University (on the left) and J. M. Wills, University Siegen (on the right). 4 Art Discrete Appl. Math. 1 (2018) #P1.02 of a regular map (3, 10) of genus 6 of W. Dyck with 15 vertices, [7]. See also a survey article of U. Brehm and E. Schulte in [8] and the papers cited there. Figure 3: Symmetric realization of Dyck’s regular map of genus 3 by U. Brehm [6]. 2 Combinatorial description For general descriptions of combinatorial regular maps we refer the reader to [9, 11], and [23]. Hurwitz’s regular map of genus 7 consists of the following 168 triangles in Table 1. It has 252 edges and 72 vertices labeled 1, . . . , 72. Compared with previous polyhedral realizations of regular maps we are faced with additional complexity. From the automorphism group of this manifold of order 1 008, we have used a dihedral subgroup of order 14 for sorting the triangles. The cyclic subgroup has the following generator. (1) (72) (2, . . . , 8) (9, . . . , 15) (16, . . . , 22) (23, . . . , 29) (30, . . . , 36) (37, . . . , 43) (44, . . . , 50) (51, . . . , 57) (58, . . . , 64) (65, . . . , 71) When we assume the vertices from 2 to 71 to form ten regular seven-gons in horizontal equidistant planes with heights sorted according to labels belonging to the same orbit of the cyclic group, we have an additional up-side-down symmetry that maps vertex 1 to vertex 72 which is an automorphism of the map. The combinatorial description of a complete list of small regular maps has been given in [10], an even more extended list is available from the first author of the same article. 3 Topological visualization The study of topological visualizations of regular maps has recently been done by J. J. van Wijk, [21, 22], by C. H. Séquin, [20], and by Razafindrazaka and Polthier, [15]. From J. J. van Wijk we have a nice topological visualization as a computer film of our Hur- J. Bokowski and M. Cuntz: Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization 5 Table 1: Triangles of Hurwitz’s surface of genus seven. (01, 02, 03), (01, 03, 04), (01, 04, 05), (01, 05, 06), (01, 06, 07), (01, 07, 08), (01, 08, 02), (02, 08, 09), (02, 10, 03), (03, 11, 04), (12, 05, 04), (13, 06, 05), (14, 07, 06), (08, 07, 15), (02, 09, 23), (03, 10, 24), (11, 25, 04), (12, 26, 05), (13, 27, 06), (14, 28, 07), (08, 15, 29), (02, 16, 10), (03, 17, 11), (12, 04, 18), (13, 05, 19), (20, 14, 06), (21, 15, 07), (22, 09, 08), (02, 23, 16), (03, 24, 17), (04, 25, 18), (19, 05, 26), (20, 06, 27), (21, 07, 28], (22, 08, 29), (22, 51, 09), (16, 52, 10), (11, 17, 53), (12, 18, 54), (55, 13, 19), (56, 14, 20), (21, 57, 15), (46, 23, 09), (47, 24, 10), (48, 25, 11), (49, 26, 12), (50, 27, 13), (44, 28, 14), (45, 29, 15), (46, 09, 32), (47, 10, 33), (48, 11, 34), (49, 12, 35), (50, 13, 36), (44, 14, 30), (45, 15, 31), (32, 09, 51), (10, 52, 33), (11, 53, 34), (12, 54, 35), (55, 36, 13), (30, 14, 56), (31, 15, 57), (36, 16, 23), (30, 17, 24), (31, 18, 25), (32, 19, 26), (20, 27, 33), (21, 28, 34), (22, 29, 35), (36, 42, 16), (30, 43, 17), (31, 37, 18), (38, 19, 32), (39, 20, 33), (21, 34, 40), (22, 35, 41), (58, 16, 42), (59, 17, 43), (37, 60, 18), (38, 61, 19), (39, 62, 20), (21, 40, 63), (64, 22, 41), (58, 52, 16), (59, 53, 17), (54, 18, 60), (55, 19, 61), (56, 20, 62), (21, 63, 57), (64, 51, 22), (36, 23, 37), (30, 24, 38), (39, 31, 25), (40, 32, 26), (41, 33, 27), (34, 28, 42), (35, 29, 43), (37, 23, 60), (38, 24, 61), (39, 25, 62), (40, 26, 63), (64, 41, 27), (58, 42, 28), (59, 43, 29), (46, 60, 23), (47, 61, 24), (48, 62, 25), (49, 63, 26), (50, 64, 27), (44, 58, 28), (45, 59, 29), (44, 30, 38), (45, 31, 39), (46, 32, 40), (47, 33, 41), (48, 34, 42), (49, 35, 43), (50, 36, 37), (30, 56, 43), (31, 57, 37), (38, 32, 51), (39, 33, 52), (40, 34, 53), (54, 41, 35), (55, 42 ,36), (50, 37, 57), (44, 38, 51), (45, 39, 52), (46, 40, 53), (47, 41, 54), (48, 42, 55), (49, 43, 56), (44, 51, 65), (45, 52, 66), (46, 53, 67), (47, 54, 68), (48, 55, 69), (49, 56, 70), (50, 57, 71), (44, 65, 58), (45, 66, 59), (46, 67, 60), (47, 68, 61), (48, 69, 62), (49, 70, 63), (50, 71, 64), (64, 65, 51), (58, 66, 52), (67, 53, 59), (68, 54, 60), (55, 61, 69), (56, 62, 70), (63, 71, 57), (58, 65, 66), (67, 59, 66), (67, 68, 60), (68, 69, 61), (69, 70, 62), (63, 70, 71), (64, 71, 65), (66, 65, 72), (67, 66, 72), (67, 72, 68), (68, 72, 69), (69, 72, 70), (71, 70, 72), (65, 71, 72). 6 Art Discrete Appl. Math. 1 (2018) #P1.02 witz surface of genus 7 in [22]. We show corresponding pictures with a dihedral symmetry D7 of order 14 in Figure 4 and Figure 5. Figure 4: Topological visualization of Hurwitz’s regular map (3, 7) of genus 7 of J. J. van Wijk, [21]. Figure 5: Topological visualization of Hurwitz’s regular map (3, 7) of genus 7 of C. H. Séquin, [20], see also [19]. Unfortunately, the method of Razafindrazaka and Polthier did not work in the case of Hurwitz’s surface of genus 7 to provide an additional topological visualization. However, we have an additional different topological visualization as a 3D-Model that was helpful during our investigation for finding a polyhedral realization, see Figure 6. J. Bokowski and M. Cuntz: Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization 7 This model of Figure 6 shows seven six-gons (marked by white connections inside the outer torus) around the axis that is fixed under the dihedral symmetry. Figure 6: Topological visualization of Hurwitz’s regular map (3, 7) of genus 7 as a 3D- model. (This model was presented at the Jörgshop at the Technical University Berlin in June 2017.) When we cut the model along those six-gons, we see that we can split the surface in two parts having 84 triangles each. On the one hand we obtain a topological torus with these seven six-gons as holes and on the other hand we have a topological 2-sphere with these seven holes. Figure 7: Triangles of the torus with seven holes each bounded by a polygon of length 6. In Figure 7 we have depicted the combinatorial torus structure and in Figure 8 we see the corresponding 84 triangles of the sphere. Whereas both of these parts of the Hur- 8 Art Discrete Appl. Math. 1 (2018) #P1.02 witz surface of genus 7 can easily be represented with planar triangles and without self- intersections, we see that the cyclic sequences of the holes in both cases do not coincide. However, they have to match. This is a clear indication that we probably cannot hope for a corresponding symmetric realization of order 7. Figure 8: Triangles of the sphere with seven holes each bounded by a polygon of length 6. 4 Polyhedral realization 4.1 An algorithm We use the following simple algorithm to obtain realizations within a few minutes (depend- ing on the choice of distances and parameters): 1. Choose randomly a set of 72 distinct points P = {P1, . . . , P72} ✓ Q3 with rational coordinates. 2. Count the number w0 of pairs of labels of a triangle and labels of an edge of a triangle in Table 1 for which the corresponding points in P produce an intersection of a triangle and an edge. 3. Remember the points involved in these w0 intersections in a set I . 4. While w0 > 0, do: (a) Move a randomly chosen point of I into a random direction in such a way that it does not go too far away and not too close to the other points. (b) As above, count the number w of intersections and remember the points in- volved in intersections in a new set I . (c) If w > w0 then move the point back to its place, else: w0 w. 5. Output the solution. J. Bokowski and M. Cuntz: Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization 9 An implementation in C produces for example the solution displayed in Table 2. To be completely sure that this output is correct one may check it using the code in Figure 9 or Figure 10. Table 2: Coordinates of a polyhedral realization without self-intersections of Hurwitz’s surface of genus seven. no. x y z no. x y z no. x y z 1 430 -270 -1000 2 959 -237 -213 3 434 -984 -70 4 -418 -861 -677 5 -988 98 -665 6 -272 -139 -814 7 299 577 -988 8 999 399 -854 9 981 727 -246 10 475 -498 408 11 361 -806 840 12 -509 115 609 13 -541, -105 26 14 -299 434 -801 15 456 -230 -780 16 819 353 803 17 841 -663 868 18 -941 982 856 19 -928 694 -73 20 21 -294 158 21 -132 450 -319 22 526 305 -430 23 782 -550 996 24 172 -288 93 25 -859 -989 528 26 -679 983 697 27 -95 -239 -217 28 764 665 653 29 563 490 169 30 -872 507 -510 31 -413 -817 -561 32 136 921 30 33 432 -176 -157 34 522 778 359 35 489 -85 120 36 -470 84 709 37 -520 -823 679 38 -383 876 -325 39 365 -758 -25 40 114 900 838 41 240 176 -191 42 234 26 700 43 4 -150 345 44 261 843 -15 45 850 19 -196 46 902 679 797 47 17 36 114 48 -331 -763 720 49 -523 632 368 50 -254 -694 -243 51 367 659 -796 52 791 -11 367 53 194 442 411 54 34 376 304 55 -132 -413 773 56 103 -743 654 57 -240 -160 -832 58 940 632 175 59 567 -43 515 60 224 233 981 61 -254 268 182 62 -271 -721 265 63 -60 540 192 64 280 -119 -630 65 644 565 -266 66 516 538 265 67 -117 524 443 68 210 227 -110 69 -275 -204 444 70 -157 44 359 71 199 402 -282 72 90 510 140 4.2 Explanations It is already very difficult to describe the geometric shape of any of the two parts of the Hurwitz surface of genus 7 that we have described in the last section. The Blender soft- ware is a powerful tool for 3D objects. In Figure 11, Figure 12, Figure 13, and Figure 14 you see some pictures of our realization. The reader can get a better understanding by using our corresponding Blender files for rotating the objects. Please write an e-mail to the authors. Glueing properly both parts along their boundaries leads to our polyhedral realization without self-intersections. How did we check that the polyhedron has no self-intersections? We first confirmed that all vertices are in general position. This is equivalent to the fact that all determinants of any 4 points (by using homogeneous coordinates) are non-zero. Afterwards we have checked all pairs (edge, triangle) for intersections. Edges that have a vertex in common with a triangle cause no problem, because the points are in general position. The other cases (edge, triangle) depend on the signs of the five determinants obtained from the five 4-element subsets of the set formed by the vertices of the edge and the trian- 10 Art Discrete Appl. Math. 1 (2018) #P1.02 gle (again using homogeneous coordinates). When the two vertices of an edge lie on the same side of the plane determined by the triangle, we have no intersection. Otherwise we pick a vertex of the edge as the apex of a cone generated by the triangle. Precisely, when all three planes determined by the faces of this cone have the other vertex of the edge on the same side as the remaining vertex of the triangle, we have an intersection. In other words the other vertex of the edge lies within the convex cone, however beyond the triangle seen from the apex. We have double checked this result with two different programming methods, Haskell and Magma. When using exploded views, corresponding films, a sym- metric realization, or even a geometric model, the reader might gain additional insight. Our attempts to find a symmetric realization were not successful. For a cyclic symmetry of order 7 we have even seen an argument that tells us how unlikely the existence of such a realization might be. coordinates:= . . . triangles:= . . . edges:=[Sort(SetToSequence(k)) : k in &join [{{a[1],a[2]},{a[1],a[3]},{a[2],a[3]}} : a in triangles]]; for e in edges, t in triangles do if #(SequenceToSet(e) meet SequenceToSet(t)) eq 0 then x:=t[1]; y:=t[2]; z:=t[3]; a:=e[1]; b:=e[2]; Y:=<[x,y,z,a],[x,y,z,b],[a,x,y,b],[a,x,y,z],[a,y,z,b],[a,y,z,x],[a,x,z,b],[a,x,z,y]>; D:=[Determinant(Matrix(4,&cat [coordinates[i] : i in u])) : u in Y]; D:=[d eq 0 select 0 else (d gt 0 select 1 else -1) : d in D]; if D[1] ne D[2] and D[3] eq D[4] and D[5] eq D[6] and D[7] eq D[8] then printf "edge %o and triangle %o: ",e,t; error "intersection!"; end if; if 0 in D then error "zero determinant!"; end if; end if; end for; Figure 9: Magma code. J. Bokowski and M. Cuntz: Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization 11 module Hurwitz where import Data.List type MA =[[Integer]] -- matrix type OB =(Tu,Or) -- oriented base type Tu = [Int] -- tuple of elements type Or = Int -- orientation -- ch=[([a,b,c,x],s),([a,b,c,y],s),([a,b,x,y],s),([a,c,x,y],s),([b,c,x,y],s)] check::[Int]->[Int]->[[Integer]]->Bool check triangle edge matrix | (length (nub(triangle++edge)) < 5) = True | snd(ch!!0) == snd(ch!!1) = True | snd(ch!!0) == snd(ch!!2) = True | snd(ch!!0) /= snd(ch!!3) = True | snd(ch!!0) == snd(ch!!4) = True | otherwise = False where ma = subMA (triangle++edge) matrix ch = m2Chi ma edges::[[Int]] edges = nub( [[el!!0]++[el!!1]|el<-triangles]++[[el!!1]++[el!!2]|el<-triangles] ++[[el!!2]++[el!!0]|el<-triangles]) tuples::Int->Int->[[Int]] -- r -> n -> all r-tuples of [1..n] tuples 0 n = [[]] tuples r n = tuplesL r [1..n] tuplesL::Int->[Int]->[[Int]]-- r -> list -> all r-tuples of list tuplesL r list@(x:xs) | length list < r = [] | length list == r = [list] | r == 1 = [[el]|el<-list] | otherwise = [[x]++el| el<-tuplesL (r-1) xs]++tuplesL r xs det::MA -> Integer -- matrix -> determinant of matrix det m |n == 1 = head (head m) |otherwise=sum(map (\i->((-1)^(i+1))⇤(head(m!!i)) ⇤(det [(map tail m)!!l|l<-[0..n-1],l/=i]))[0..n-1]) where n = length m dets::[[Int]]->MA-> [Integer]-- rsets -> matrix -> (r x r)-sub-determinants dets sets matrix = [det[matrix!!(i-1)|i<-set]|set<-sets] m2Chi::MA->[OB] -- matrix -> chirotope of matrix m2Chi m =zip trn (map fromInteger (map signum(dets trn m))) where n = length m r = length(head m) trn= tuples r n subMA::[Int]->MA->MA -- indices -> matrix -> submatrix subMA t m = map(\i->m!!(i-1))t coord::MA coord = list of homogeneous coordinates triangles::[[Int]] triangles= list of triangles Figure 10: Haskell code with some explanations. 12 Art Discrete Appl. Math. 1 (2018) #P1.02 Figure 11: Polyhedral realization of the sphere with seven holes each bounded by a polygon of length 6. J. Bokowski and M. Cuntz: Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization 13 Figure 12: Polyhedral realization of the torus with seven holes each bounded by a polygon of length 6. Three orthogonal projections and a perspective view. Even half of the complete polyhedral realization is difficult to understand. 14 Art Discrete Appl. Math. 1 (2018) #P1.02 Figure 13: Polyhedral realization Hurwitz’s surface of genus 7. J. Bokowski and M. Cuntz: Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization 15 Figure 14: Polyhedral realization Hurwitz’s surface of genus 7, complete wireframe. 16 Art Discrete Appl. Math. 1 (2018) #P1.02 References [1] J. Bokowski, Aspects of computational synthetic geometry II: Combinatorial complexes and their geometric realization — An algorithmic approach, in: H. Crapo (ed.), Computer-Aided Geometric Reasoning, Institut National de Recherche en Informatique et en Automatique (IN- RIA), Antibes, France, 1987 pp. 87–125, proceedings of the INRIA Workshop (Sophia Antipo- lis, 1987). [2] J. Bokowski, A geometric realization without self-intersections does exist for Dyck’s regular map, Discrete Comput. Geom. 4 (1989), 583–589, doi:10.1007/bf02187748. [3] J. Bokowski, On the geometric flat embedding of abstract complexes with symmetries, in: K. H. Hofmann and R. Wille (eds.), Symmetry of Discrete Mathematical Structures and Their Symmetry Groups, Heldermann, Berlin, volume 15 of Research and Exposition in Mathematics, 1991 pp. 1–48, a collection of essays. [4] J. Bokowski and B. Sturmfels, Computational Synthetic Geometry, volume 1355 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1989, doi:10.1007/bfb0089253. [5] J. Bokowski and J. M. Wills, Regular polyhedra with hidden symmetries, Math. Intell. 10 (1988), 27–32, doi:10.1007/bf03023848. [6] U. Brehm, Maximally symmetric polyhedral realizations of Dyck’s regular map, Mathematika 34 (1987), 229–236, doi:10.1112/s0025579300013474. [7] U. Brehm and U. Leopold, A symmetric combinatorially regular polyhedron of genus 6 with 15 vertices, in preparation. [8] U. Brehm and E. Schulte, Polyhedral maps, in: J. E. Goodman and J. O’Rourke (eds.), Hand- book of Discrete and Computational Geometry, CRC Press, Boca Raton, Florida, CRC Press Series on Discrete Mathematics and its Applications, 1997 pp. 345–358. [9] M. Conder, Orientable regular maps of genus 2 to 101, 2006, http://www.math. auckland.ac.nz/~conder/OrientableRegularMaps101.txt. [10] M. Conder and P. Dobcsányi, Determination of all regular maps of small genus, J. Comb. Theory Ser. B 81 (2001), 224–242, doi:10.1006/jctb.2000.2008. [11] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 4th edition, 1980, doi:10.1007/978-3-662-21943-0. [12] A. Hurwitz, Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403–442, doi:10.1007/bf01443420. [13] A. M. Macbeath, On a curve of genus 7, Proc. London Math. Soc. 15 (1965), 527–542, doi: 10.1112/plms/s3-15.1.527. [14] P. Mani, Automorphismen von polyedrischen Graphen, Math. Ann. 192 (1971), 279–303, doi: 10.1007/bf02075357. [15] F. Razafindrazaka and K. Polthier, Regular surfaces and regular maps, in: G. Green- field, G. Hart and R. Sarhangi (eds.), Proceedings of Bridges 2014: Mathemat- ics, Music, Art, Architecture, Culture (Seoul, Korea, 2014), Tessellations Publishing, Phoenix, Arizona, 2014 pp. 225–234, http://archive.bridgesmathart.org/ 2014/bridges2014-225.html. [16] E. Schulte and J. M. Wills, A polyhedral realization of Felix Klein’s map {3, 7}8 on a Riemann surface of genus 3, J. London Math. Soc. 32 (1985), 539–547, doi:10.1112/jlms/s2-32.3.539. [17] E. Schulte and J. M. Wills, Combinatorially regular polyhedra in three-space, in: K. H. Hof- mann and R. Wille (eds.), Symmetry of Discrete Mathematical Structures and Their Symmetry J. Bokowski and M. Cuntz: Hurwitz’s regular map (3, 7) of genus 7: A polyhedral realization 17 Groups, Heldermann, Berlin, volume 15 of Research and Exposition in Mathematics, 1991 pp. 49–88, a collection of essays. [18] E. Schulte and J. M. Wills, Convex-faced combinatorially regular polyhedra of small genus, Symmetry 4 (2012), 1–14, doi:10.3390/sym4010001. [19] C. H. Séquin, My search for symmetrical embeddings of regular maps, in: G. W. Hart and R. Sarhangi (eds.), Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (Pécs, Hungary, 2010), Tessellations Publishing, Phoenix, Arizona, 2010 pp. 85–94, http://archive.bridgesmathart.org/2010/bridges2010-85.html. [20] C. H. Séquin, private communication to J. J. van Wijk, 2010. [21] J. J. van Wijk, Symmetric tiling of closed surfaces: visualization of regular maps, ACM Trans. Graph. 28 (2009), 49:1–49:12, doi:10.1145/1531326.1531355. [22] J. J. van Wijk, Visualization of regular maps: the chase continues, IEEE Trans. Vis. Comput. Graphics 20 (2014), 2614–2623, doi:10.1109/tvcg.2014.2352952. [23] A. Vince, Maps, in: J. L. Gross and J. Yellen (eds.), Handbook of Graph Theory, CRC Press, Boca Raton, Florida, Discrete Mathematics and its Applications, 2004. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.03 https://doi.org/10.26493/2590-9770.1218.dac (Also available at http://adam-journal.eu) Classification of robust cycle bases and relations to fundamental cycle bases Alexander Reich Mathematics Institute, Brandenburg University of Technology, Platz der Deutschen Einheit 1, D-03046 Cottbus, Germany Received 29 September 2016, accepted 15 September 2017, published online 10 November 2017 Abstract The construction of a cycle in a graph can be realized by iteratively adding cycles of a cycle basis. The construction of each elementary cycle is only possible if this cycle basis is robust. In the last years, different classes of robust cycle bases have been established. We compare these classes and show that they are completely unrelated. More precisely, we draw a Venn diagram which displays the obvious containedness relations and show that each of its regions is not empty. In addition, we continue the comparison with fundamental cycle bases. Keywords: Minimum cycle basis, robust cycle basis, quasi-robust cycle basis, fundamental cycle basis. Math. Subj. Class.: 05C10, 05C38, 05C50 1 Introduction Cycle bases of graphs have numerous applications, e.g. in the fields of periodic timetable optimization [9], coordination of traffic signals [15], or chemistry [4]. The first reference [9] additionally provides a useful classification of several types of cycle bases utilized for computations in the mentioned areas. The author considered the seven classes of directed, undirected, integral, totally unimodular, planar, as well as weakly and strictly fundamental cycle bases and compared them to each other. Another line of research has been initiated by Kainen [6] who investigated robust cycle bases. Strengthening and weakening the concept of robust cycle bases led to four different types of robust cycle bases, which were further studied in [8] and [12], and recently in [7]. The latter paper provides an application of robust cycle bases to the analysis of commutative diagrams in groupoids. E-mail address: alexanderreich@arcor.de (Alexander Reich) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.03 Similarly to the work of Liebchen [9], we show in our paper that no two of these four classes coincide and give a separating example for each pair of the classes. All of our examples provide a graph with its uniquely minimum cycle basis. This indicates that each class of robust cycle bases admits its own minimization problem. A further focus in this paper is the relationship of robust and fundamental cycle bases, The investigation on this topic has been initiated in [8]. We continue this research by providing more examples of cycle bases which are even minimum in almost all cases. We are able to eliminate one of two question marks in a map given there, where the authors conjectured the existence of examples. The results in this paper appeared also in the thesis [13]. 2 Preliminaries Throughout the paper, we consider only simple undirected weighted graphs G = (V,E) with finite node set V (G) = V, finite edge set E(G) = E, and weight function w : E ! R>0. The degree of a node v 2 V is denoted by deg(v). A path P of length ` in a graph is a sequence P = (v0, v1, . . . , v`) of pairwise disjoint nodes with vi1vi 2 E for 1  i  `. The length of a shortest path between two nodes u and v in G is called the distance distG(u, v). A path from node u to node v is referred to as u-v-path. A circuit C in G is a non-empty connected subgraph of G with deg(v) = 2 for all v 2 V (C). We define |C| := P e2E(C) w(e) as the length of a circuit C. A cycle Z in G is a subgraph of G where deg(v) is even for all v 2 V (Z). For a spanning tree T = (V,E(T )) of G and an edge e 2 E \ E(T ) define the funda- mental circuit CT (e) as the unique circuit in (V,E(T )[ {e}). The non-tree edges are also called chords of the spanning tree T . We usually identify circuits, cycles, and trees with their edge sets. The cycle space C(G) of a graph G = (V,E) is the vector subspace of GF(2)E that is generated by the incidence vectors of the circuits in G. The sum of two cycles Z1 and Z2 in this vector space is their symmetric difference (Z1 [ Z2) \ (Z1 \ Z2). A cycle basis B of G is a set of ⌫ = m n + 1 circuits whose incidence vectors form a basis of C(G). The size (B) of a cycle basis B is defined as (B) := P C2B |C|. A cycle basis B of G = (V,E) is designated strictly fundamental iff there is a spanning tree T = (V,E(T )) with B = {CT (e) | e 2 E \ E(T )}. A cycle basis B = {Z1, . . . , Z⌫} is weakly fundamental if there exists a permutation ⇡ 2 S⌫ such that Z⇡(i) \ i1[ j=1 Z⇡(j) 6= ; for all i = 2, . . . , ⌫. (2.1) Weakly fundamental cycle were also the matter of [5] where the authors characterized graphs for which every cycle basis is weakly fundamental. If B is a cycle basis then every cycle Z has a unique representation Z = P C2B CC with C 2 {0, 1}. The subset {C 2 B | C = 1 and Z = P C2B CC} is called the support supp(Z). The following simple lemma is needed to justify the minimality of some of our cycle bases. Lemma 2.1. For a given strictly fundamental cycle basis B of an undirected graph G = (V,E) one can always find a weight function w such that B is the unique minimum cycle A. Reich: Classification of robust cycle bases and relations to fundamental cycle bases 3 basis of G. Proof. Let T be a fundamental spanning tree which induces B. For every edge e 2 T set w(e) = 1. Define d := max{distT (u, v) | uv 2 E\T} and assign w(e) = 2ddistT (u, v) for the remaining edges e = uv. Observe that the minimum of w restricted to the chords is d. Now, every circuit in B has a weight of 2d while all other cycles of G have a greater weight since they contain at least two chords and at least one tree edge or at least three chords. For Example 5.5, the following enhancement of Lemma 2.1 is necessary. Lemma 2.2. For a given strictly fundamental cycle basis B of an undirected graph G = (V,E) one can always find a weight function w such that B is the unique minimum cycle basis of G and such that there is a chord e = uv with w(e) < distT (u, v). Proof. The proof has essentially the same structure as the proof of Lemma 2.1. Thus, set w(e) = 1 for all tree edges of a given fundamental spanning tree T which induces B. And again, let d := max{distT (u, v) | uv 2 E \ T}. For the edges e = uv in E \ T , we now assign the weight w(e) = 2d distT (u, v) ", for an " > 0 whose value is determined later. The minimum of w restricted to the chords is d ", and each circuit C 2 B has the weight w(C) = 2d ". Now, look at a circuit which is not in B. It consists of c 2 chords and t 0 tree edges. Furthermore, c = 2 implies t 1. The length of the circuit is at least c(d ") + t. For all " 2 (0, (c2)d+tc1 ), this value is greater than 2d ", i.e. greater than the weight of a basic circuit. Because c 2, the denominator of the upper endpoint of the interval is not zero, and since c+ t 3, also the numerator is not zero. Hence, this interval is not empty and we can take any " from this interval. Finally, for a chord e = uv with distT (u, v) = d, the weight w(e) has the value d " < distT (u, v). 3 Classes of robust cycle bases In order to define the four different types of robust cycle bases, we essentially follow the exposition in [12]. Similarly as there, we need at first the concept of (strictly) well-arranged sequences of circuits. Afterwards, we deduce several simple inclusions and present a map of the relationship between the different classes of robust cycle bases. Definition 3.1 ((Strictly) well-arranged sequence). A sequence S = (C1, . . . , Ck) of cir- cuits in an undirected graph is called well-arranged if for all j = 1, . . . , k the GF(2)-sumPj i=1 Ci is also a circuit. A well-arranged sequence of circuits is strictly well-arranged if for all j = 2, . . . , k the intersection Cj \ Pj1 i=1 Ci is a single path. The path in Definition 3.1 contains at least one edge. Otherwise, the sum Cj+ Pj1 i=1 Ci was not a circuit and thus, the sequence was not even well-arranged, at all. It is clear that every strictly well-arranged sequence is also well-arranged. Furthermore, it is known that there are well-arranged sequences that are not strictly well-arranged. The authors of [8] provide such an example in which the sum of two basic circuits is again a circuit, but they intersect in three paths. Note that it is not forbidden that a circuit appears more than once in a (strictly) well-arranged sequence. 4 Art Discrete Appl. Math. 1 (2018) #P1.03 With this in mind, we are now able to define the four different types of robust cycle bases which were developed in [12]. Definition 3.2 (Cyclically/strictly robust and (strictly) quasi-robust cycle basis). A cycle basis B of a graph G is (strictly) quasi-robust if for each circuit C in G there is a (strictly) well-arranged sequence SC = (C1, . . . , Ck1, Ck) such that C = Pk i=1 Ci and Ci 2 B for i = 1, . . . , k. A strictly quasi-robust cycle basis is strictly robust if the circuits in the strictly well-arranged sequence are pairwise disjoint. Analogously, a quasi-robust cycle basis is cyclically robust if the according well-arranged sequence does not contain a circuit twice. If we do not want to specify the particular type of robustness, we simply speak about a robust cycle basis. It can be concluded that for strictly and for cyclically robust cycle bases the well- arranged sequence of a circuit C must not contain basic circuits which are not in the support of C. Also, directly from these definitions, we can immediately derive the following facts: • every strictly quasi-robust cycle basis is quasi-robust, • every strictly robust cycle basis is strictly quasi-robust, • every strictly robust cycle basis is cyclically robust, and • every cyclically robust cycle basis is quasi-robust. These inclusions hold since in each case, we require additional properties for the more specific class. The inclusions give rise to the diagram in Figure 1. cyclically robust robust strictly strictly quasi-robust quasi-robust not quasi-robust Ex. 4.1 Ex. 4.2 Ex. 4.3 Ex. 4.4 Ex. 4.5Ex. 4.6 Figure 1: Map of robust cycle bases. Not much is known about which graph classes can have which type of robust cycle bases. Furthermore, it is unknown whether each graph admits a robust cycle basis of any of the four types. Table 1 summarizes the related results. To the best of our knowledge, these are the only known ones. A. Reich: Classification of robust cycle bases and relations to fundamental cycle bases 5 Table 1: Summary of known graph classes for which the stated type of cycle bases is guaranteed. Graph class Robustness Reference planar graphs strictly robust [2] complete graphs strictly robust [6] complete bipartite graphs Km,n with m  4 and n  5 strictly robust [12] general complete bipartite graphs quasi-robust [12] wheels cyclically robust [8] 4 Examples of robust cycle bases In this section, we show that the inclusions derived in the last section are valid only in the given direction. Thus, no two of the classes are equivalent. To point this out, we give an example of a graph and a cycle basis for each region in the map in Figure 1 and thus show that it is not empty. Except in Example 4.2, all cycle bases are strictly fundamental. According to Lem- ma 2.1, we can choose a weight function such that this cycle basis is the unique minimum cycle basis on this graph. However, the given cycle basis in Example 4.2 is also the unique minimum one. The existence of a graph with a minimum cycle basis in each region of the map indicates that each class—actually even each non-empty difference of two classes—of robust cycle bases admits its own minimization problem. Remember that we do not know an efficient algorithm for the computation or for the recognition of any type of robust cycle bases on general graphs. Thus, to prove a cycle basis of a graph G as (strictly) quasi-robust, we have to indicate a (strictly) well-arranged sequence of basic circuits for every circuit in G. Analogous sequences have to be found for (strictly) robust cycle bases. In the latter case, a basic circuit is allowed to occur at most one time in each of these sequences. On the other hand, a cycle basis B of a graph G is not quasi-robust if there exists a circuit C 0 in G such that for each C 2 B the sum C 0 + C is not a circuit. To show that the cycle basis is not strictly quasi-robust, one has to verify that the cut C 0 \ C does not form a path for one circuit C 0 in the graph and for all C 2 B. Finally, to show that a cycle basis is not a cyclically robust or a strictly robust cycle basis it suffices to check only the circuits of the support of such a circuit C 0. We now start with the description of the examples. Example 4.1 (Strictly robust cycle basis). The first example is the simple graph C3 that consists of exactly one circuit of length 3. Clearly, its unique cycle basis is strictly robust— and strictly fundamental and minimum, as well. Example 4.2 (Cyclically robust and strictly quasi-robust cycle basis—not strictly robust). Our second example is the complete bipartite graph K3,3, see Figure 2 (a). The cycle basis B = {C1, C2, C3, C4} is highlighted in Figure 2 (b). It is not strictly fundamental, thus, we suggest the indicated weights to make the cycle basis minimum. All other circuits have a greater weight. The weights of all circuits are denoted below the graph in Figures 2 (b), (c) and (d). We show that B is cyclically robust and strictly quasi-robust, but not strictly robust. For 2  k  ⌫ we denote Ci1,...,ik := Pk j=1 Cij . 6 Art Discrete Appl. Math. 1 (2018) #P1.03 1 1111 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 22 2 2 2 2 2 2 22 2 2 2 22 2 2 2 2 2 2 2 3 33 33 3 3 3 33 33 33 3 3 3 33 3 3 3 3 3 3 3 3 K3,3 C1 C2 C3 C4 C1,2 C1,3 C1,4 C2,3 C2,4 C3,4 C1,2,3 C1,2,4 C1,3,4 C2,3,4 C1,2,3,4 C1,2,3 ∩ C1 C1,2,3 ∩ C2 C1,2,3 ∩ C3 7777 88 9 9 10 10 11 121212 15 (a) (b) (c) (d) (e) Figure 2: The K3,3 with weights on the edges (a). The four basic circuits and their weights below (b). All other circuits and their weights (c) and (d). The intersections (dashed edges) of C1,2,3 with the circuits of its support (e). Cyclically robust. The K3,3 is cubic, hence it contains no cycle with vertices of degree 4 or more. Furthermore, it has only six vertices, but it is triangle free. Thus, there is no cycle consisting of two triangles. This means that every cycle is a circuit and therefore, each cycle basis of K3,3 is cyclically robust. Not strictly robust. The given basis is not strictly robust, since there is no strictly well- arranged sequence for C1,2,3, in which every basic circuit occurs only once. Observe this by looking at Figure 2 (e). It is indicated that C1,2,3 has an intersection consisting of two path (dashed edges) with each circuit from its support. Strictly quasi-robust. For the circuits which have exactly two basic circuits in their supports, these two circuits intersect in a single path, illustrated by the dashed edges in Figure 2 (c). For the circuits depicted in Figure 2 (d) we provide the sequences SC1,2,3 = (C1, C3, C4, C2, C4), SC1,2,4 = (C1, C4, C2), SC1,3,4 = (C1, C3, C4), SC2,3,4 = (C2, C4, C3), and SC1,2,3,4 = (C1, C3, C4, C2), which are all strictly well-arranged. Hence, the cycle basis is strictly quasi-robust. Example 4.3 (Quasi-robust cycle basis—neither cyclically robust nor strictly quasi-ro- bust). This example is borrowed from [6]. We consider the complete bipartite graph K5,5, the strictly fundamental cycle basis B induced by the spanning tree T shown in Figure 3 (a), and the circuit C aside in Figure 3 (b). The sixteen basic circuits themselves are also de- A. Reich: Classification of robust cycle bases and relations to fundamental cycle bases 7 picted in Figure 4 as black edges. Assigning weights according to Lemma 2.1, B becomes the unique minimum cycle basis. T C (a) (b) Figure 3: Spanning tree T of K5,5 (a). The circuit C considered in the text (b). (a) C1 C2 C3 C4 C5 C6 16 (b) C7 C8 C9 C10 C11 C12 16 (c) C13 C14 C15 C16 Figure 4: The sixteen basic circuits of B (black edges) and the circuit C (grey edges). For the sake of clearness we dropped the edges which are neither in the basic circuit nor in C. Quasi-robust. The described basis had been shown to be quasi-robust in [12] in an elaborate manner. Not cyclically robust. The circuit C can be written as C = P6 i=1 Ci and all the sums C + Ci for i = 1, . . . , 6 are cycles with node degrees greater than 2 (marked by a circle), see Figure 4 (a). Hence, this cycle basis is not cyclically robust. Not strictly quasi-robust. Looking at the remaining basic circuits, we observe that also C7 to C12 yield cycles with node degrees of 4, Figure 4 (b), again marked by a circle. The intersection of C13 to C16 with C is not a single path in each case, as can be seen in Figure 4 (c). In addition, C+C13 and C+C14 are disconnected. All in all, the cycle basis is not strictly quasi-robust. Example 4.4 (Cycle basis, not even quasi-robust). The example of a cycle basis which is 8 Art Discrete Appl. Math. 1 (2018) #P1.03 not even quasi-robust presented here had been inspired by a talk of Ostermeier [11].1 The cycle basis is strictly fundamental and it is induced by the fat drawn tree in Fig- ure 5 (a). (a) (b) (c) e1 e2 C Figure 5: Graph with an inducing fundamental spanning tree (fat edges) and dashed chords e1 and e2 (a), a circuit C (b), and sums of C with two basic circuits generated by the chords e1 and e2 (c). Not quasi-robust. Due to symmetry, we have to consider only the basic circuits induced by the dashed edges e1 and e2. In both cases, they add up with C to a cycle that is not a circuit, see Figure 5 (c). Example 4.5 (Cyclically robust cycle basis—not strictly quasi-robust). This example is a cycle basis on Wagner’s graph V8 which is cyclically robust, but not strictly quasi-robust. The strictly fundamental basis is indicated by the spanning tree which is highlighted in Figure 6 (a). The basic circuits are denoted at the chords. We use the notation from Exam- ple 4.2, i.e. Ci1,...,ik := Pk j=1 Cij . (a) (b) (c) (d) C C1 C2 C3 C4 C5 C3,4 C1,2,5 C ∩ C1 C ∩ C2 C ∩ C3 C ∩ C4 C ∩ C5 Figure 6: Wagner’s graph V8 with a fundamental spanning tree (a). The only two non- circuits in V8 (b). The circuit C (c). The intersections of C (grey) with the five basic circuits do not form a single path (d), edges which are not in a basic circuit or in C are dropped. Cyclically robust. Wagner’s graph V8 is cubic which implies that every cycle is 2- regular. The only critical cycles in V8 are thus the two non-circuit pictured in Figure 6 (b). We provide the well-arranged sequences SC3,4+C1 = SC1,3,4 = (C1, C3, C4), SC3,4+C2 = SC2,3,4 = (C2, C3, C4), and SC3,4+C5 = SC3,4,5 = (C4, C5, C3) for the circuits which arise by adding a remaining basic circuit to C3,4. For the cycle C1,2,5 we give the sequences 1A similar example already appeared in [14]. A. Reich: Classification of robust cycle bases and relations to fundamental cycle bases 9 SC1,2,5+C3 = SC1,2,3,5 = (C1, C2, C3, C5) and SC1,2,5+C4 = SC1,2,4,5 = (C1, C2, C4, C5). In each of these sequences, every basic circuit appears at most once. This shows that the basis B is cyclically robust. Not strictly quasi-robust. To see that the basis is not strictly quasi-robust, consider the circuit C in Figure 6 (c). Its intersection with each basic circuit does not form a single path. This is illustrated in Figure 6 (d). Example 4.6 (Strictly quasi-robust cycle basis—not cyclically robust). The last example provides a graph with a cycle basis B = {C1, . . . , C6} which is strictly quasi-robust but not cyclically robust. As in Example 4.2 denote Ci1,...,ik := Pk j=1 Cij for 2  k  ⌫. C1 C2 C3 C4 C5 C6 Figure 7: Graph with a fundamental spanning tree which induces a cycle basis that is strictly quasi-robust but not cyclically robust. Strictly quasi-robust. Since ⌫ = 6 we have to investigate 26 6 1 = 57 cycles; the six basic circuits and the zero vector are not interesting. The 22 cycles listed below are not circuits. C1,4, C2,3, C1,5,6, C4,5,6, C1,4,5,6, C2,3,5,6, C1,2,3,4,5, C1,2,4,5,6, C1,5, C1,4,5, C2,4,6, C1,2,4,6, C2,3,4,5, C2,4,5,6, C1,2,3,4,6, C1,3,4,5,6, C1,6, C1,4,6, C3,4,5, C1,3,4,5, C2,3,4,6, C3,4,5,6 For the remaining eleven circuits Ci,j with | supp(Ci,j)| = 2 we may ignore the order of the basic circuits. The intersection of the two basic circuits is a path in each case, and thus, the sequences are strictly well-arranged. For the 24 circuits with at least three elements in their supports, we provide the following strictly well-arranged sequences. (C1, C2, C3), (C1, C3, C6), (C4, C6, C3), (C1, C2, C5, C6), (C1, C2, C4), (C3, C4, C2), (C5, C6, C3), (C4, C6, C3, C1), (C1, C2, C5), (C3, C5, C2), (C1, C2, C3, C4), (C5, C6, C3, C1), (C1, C2, C6), (C3, C6, C2), (C1, C2, C3, C5), (C1, C2, C3, C5, C6), (C1, C3, C4), (C4, C5, C2), (C1, C2, C3, C6), (C1, C2, C3, C6, C5, C4, C1), (C1, C3, C5), (C2, C5, C6), (C4, C5, C2, C1), (C1, C2, C3, C5, C6, C4). Not cyclically robust. Figure 8 illustrates that the treated cycle basis is not cycli- cally robust. More precisely, look at the circuit C2,3,4,5,6. For i = 2, . . . , 6, the cycles C2,3,4,5,6+Ci have nodes with degree greater than 2, marked by circles in Figure 8. Hence, this circuit does not admit a well-arranged sequence in which the circuits are pairwise dis- joint. 5 Relationship with fundamental bases One approach for a better understanding of strictly robust and cyclically robust cycle bases had been presented in [8]. Therein, the authors investigated the relationship between strictly 10 Art Discrete Appl. Math. 1 (2018) #P1.03 C2 C3 C4 C5 C6 Figure 8: The circuit C2,3,4,5,6 (grey) and the five basic circuits of its support (black edges). robust, cyclically robust, and non-robust cycle bases on one hand, and strictly fundamental, weakly fundamental, and non-fundamental cycle bases on the other hand. Their motivation was the detailed exploration of strictly and weakly fundamental cycle bases which had been done in the years before. They concluded that robustness and fundamentality of cycle bases “are essentially unrelated concepts”. In more detail, they considered the combination (robustness,fundamentality), where robustness 2 {“strictly robust”, “robust”, “non-robust”} and fundamentali- ty 2 {“strictly fundamental”, “weakly fundamental”, “non-fundamental”}. This immedi- ately led to nine possibilities, and an example of a graph with an according cycle basis was presented in seven of these cases. In this section, we follow up this line of research and provide for eight cases a graph with an appropriate cycle basis which is additionally minimum. For the ninth case, we are able to retire to a strictly quasi-robust cycle basis instead of a strictly robust one. However, this basis is not the minimum basis of the presented graph. At the end of this section, we summarize our results in Table 2. We start with three examples of strictly fundamental bases, that is the first column in Table 2. Two of them are taken from [8], the third one correlates to the basis in Example 4.5. Due to Lemma 2.1, all bases can be made minimum. Example 5.1 (Strictly fundamental—strictly robust). This example is directly taken from [8]. To be more accurate, we deal with the complete graph Kn and the cycle basis Bn which is induced by the complete bipartite graph K1,n1 as fundamental spanning tree. It is strictly robust as shown in [6]. With a weighting assigned according to Lemma 2.1 it is also the unique minimum cycle basis. We decided to present this example here because it constitutes a whole class of graphs and cycle bases with the required properties. On the other hand, also the triangle graph in Example 4.1 could have served as an example at this place. Example 5.2 (Strictly fundamental—not strictly robust—cyclically robust). Wagner’s graph V8 and the cycle basis which had already been presented in Example 4.5 provide the nec- essary properties for this example. We remark that this example eliminates one of the two question marks in [8] where the authors conjectured the existence of such an example. Example 5.3 (Strictly fundamental—not cyclically robust). Again, we borrow the example given in [8] which is called there “Ostrowski’s basis”. It is simply the K5 with a path consisting of four edges as fundamental spanning tree. This spanning tree induces a basis consisting of three triangles, two quadrangles, and one pentagon. To verify that the basis is non-robust, take a look at the circuit C which is the sum of the three triangles and the two quadrangles. The sum of C with each of these basic circuits constitutes a non-circuit. A. Reich: Classification of robust cycle bases and relations to fundamental cycle bases 11 Similarly to Example 5.2, we could have borrowed the graph with a non-robust cy- cle basis from Example 4.4. Anyway, we used Ostrowski’s basis at this place because there is an easy way to construct an infinite class of graphs and cycle bases with the re- quired properties. More precisely, we speak about the family of complete graphs with an odd number of vertices. For such a graph Gk = (Vk, Ek) with Vk = {v0, v1, . . . , v2k} we choose the path (v0, v1, . . . , v2k) as inducing spanning tree for the strictly fundamen- tal cycle basis. As a certificate for the non-robustness, we provide the circuit Ck =S2k i=0{vivi+2} = P2k i=0{vivi+1, vi+1vi+2, vi+2vi}, where the indices are taken modulo 2k + 1. Adding one basic circuit Cik = {vivi+1, vi+1vi+2, vi+2vi} to Ck results in a cycle C 0k with degC0k(vi+1) = 4. In Figure 9, the graph G3 is given as an example. v0 v1 v2 v3 v4 v5 v6 Figure 9: The graph G3, the inducing spanning tree (fat edges), and the circuit Ck (dashed edges). We continue with three examples which are weakly fundamental but not strictly fun- damental. In Table 2, these examples appear in the second column. One example is taken from [9]. For the other two, we destroy the strictly fundamentality of the according exam- ples above by gluing suitable graphs together. In doing so, we keep in mind that we want the bases to stay minimum. Example 5.4 (Not strictly fundamental—weakly fundamental—strictly robust). The addi- tional demand for a minimum cycle basis prevents us from simply copying the according example in [8]. Instead, we copy Example 11.2 from [9], which deals with the sunflower graph SF(3), depicted in Figure 10. Therein, it served as an example for a 2-basis which is not strictly fundamental, where for a 2-basis, each edge is contained in at most two basic cycles. Figure 10: The sunflower graph SF(3). The cycle basis B consisting of the only four triangles is the unique minimum cycle basis. Each edge of the middle triangle is contained in another circuit of B, thus, B is not strictly 12 Art Discrete Appl. Math. 1 (2018) #P1.03 fundamental. But since the basis is a 2-basis, it is weakly fundamental, see e.g. [9], and strictly robust due to [2]. Example 5.5 (Not strictly fundamental—weakly fundamental—not strictly robust—cycli- cally robust). The idea in this example is to adapt Wagner’s graph and its cycle basis presented in Examples 4.5 resp. 5.2 such that it is not strictly fundamental anymore. To do this, we append a further path (v1, v2, v3) at the two adjacent vertices v1 and v3 at the right hand side of the graph, see Figure 11. 1 1 6 − ε6 − ε 9 − ε 9 − ε 9 − ε v1 v2 v3 Figure 11: The modified Wagner’s graph with a partial spanning tree (fat edges) and a circuit without private edge (dashed). The weights of the graph are assigned according to Lemma 2.2. The second statement of this lemma does hold for v1v3, i.e. w(v1v3) < distT (v1, v3). For the new edges set w(v1v2) = w(v2v3) = 1. To yield the cycle basis, inherit the basic circuits from the original example and append the circuit C6 = (v1v2, v2v3, v3v1). Remark that the weights of the old edges were chosen according to Lemma 2.2 and that C6 is the shortest circuit which contains the new vertex v2. Hence, the obtained cycle basis is minimum. The basis is not strictly robust for the same reasons as in Example 4.5. On the other hand, assume that a circuit C in this graph contains the vertex v2. A well-arranged sequence for C can be achieved by concatenating C6 with the well-arranged sequence of C + C6, hence, the basis is cyclically robust. Finally, the cycle basis is not strictly fundamental, since the dashed basic circuit does not have a private edge. But it is weakly fundamental because Inequality (2.1) holds for each permutation ⇡ with C⇡(6) = C6. Example 5.6 (Not strictly fundamental—weakly fundamental—not cyclically robust). Sim- ilarly to the example above, we destroy the strictly fundamentality of Ostrowski’s basis of the K5. We also could have used Lemma 2.2 and could have constructed a graph by simply appending a path of length 2 as in Example 5.5. Anyway, we decided to provide a larger example in favor of an integer weight function. Remember that the basis of this graph was induced by a path of four edges as funda- mental spanning tree. There is one edge between the end nodes of the path, denote it eP . Now take three copies of K5 and assemble them in a way such that the three copies of eP constitute a triangle, add a vertex and connect it to the three corners of the triangle. See Figure 12 for the construction. The edge weights in the three copies of K5 are assigned according to Lemma 2.1, the three new edges get the weight 2. Again, the fat edges get weight 1. To get a cycle basis for the merged graph, combine the cycle bases of the three copies and append the three new triangles with weight 8, i.e. the triangles constituted by two new edges and one copy of eP . The ⌫ = 21 shortest circuits have weight 8, hence, the combined A. Reich: Classification of robust cycle bases and relations to fundamental cycle bases 13 2 22 4 44 5 5 5 55 5 6 6 6 6 6 6 6 6 6 Figure 12: Three merged copies of K5 with Ostrowski’s bases. cycle basis is minimum. It is not robust because Ostrowski’s basis is not. It is not strictly fundamental since the circuits induced by eP in each K5 do not have private edges, as well as the three new triangles. In the end, it is weakly fundamental. Permute the basis such that the three new triangles appear at first, followed by the three circuits induced by the copies of eP . The last three examples present non-fundamental cycle bases, listed in the third column of Table 2. Two of them are again borrowed from [9]. Example 5.7 (Not weakly fundamental—strictly quasi-robust). Unfortunately, we were not able to give an example of a minimum non-fundamental cycle basis which is strictly robust. But we provide a strictly quasi-robust one, at least. Therefore, look at the graph depicted in Figure 13 and the indicated cycle basis. (a) (b) C1 C2 C3 C4 C5 C6 Figure 13: A graph (a) and a non-fundamental cycle basis which is strictly quasi-robust, but not strictly robust (b). The basis is non-fundamental since each edge is contained in at least two basic circuits. 14 Art Discrete Appl. Math. 1 (2018) #P1.03 To see that it is strictly quasi-robust, we take a look at 26 6 1 = 57 cycles, analogous to Example 4.6. Among these cycles, there are 38 which do not constitute circuits. For the other 19 circuits, we provide the strictly well-arranged sequences below. (C1, C3), (C2, C3, C5), (C1, C3, C4, C6, C2), (C2, C3), (C6, C5, C2), (C3, C1, C4, C6, C5, C2, C3), (C4, C6), (C6, C5, C4), (C6, C5, C2, C3, C1), (C5, C6), (C1, C3, C4, C6), (C1, C3, C4, C6, C5), (C3, C2, C1), (C2, C3, C5, C6), (C2, C3, C5, C6, C4), (C1, C3, C4), (C6, C4, C1, C3, C2, C5, C6), (C6, C4, C1, C3, C2, C5) (C6, C4, C1), For the circuits which belong to the bold written sequences, there are no strictly well- arranged sequences in which all circuits are pairwise disjoint. Thus, the cycle basis is strictly quasi-robust, but not strictly robust. Example 5.8 (Not weakly fundamental—not strictly robust—cyclically robust). The cycle basis in this example is borrowed from [9] where it serves as an example of a minimum cycle basis which is not integral2. It is a basis of the generalized Petersen graph P11,4, see Figure 14. i0 i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 o0 o1 o2 o3 o4 o5 o6 o7 o8 o9 o10 Figure 14: Generalized Petersen graph P11,4 with the basic circuit C1 (dashed) and the circuit C1,4,12 = C1 + C4 + C12 (grey). The discussed basis B contains the circuits Cj+1 = (ojij , ijij+4, ij+4ij+8, ij+8ij+1, ij+1oj+1, oj+1oj) for j = 0, . . . , 10 where the indices are taken modulo 11, and the circuit C12 = {o0o1, . . . , o9o10, o10o0}. In the figure above we emphasized the circuit C1 with dashed edges. With the weights w(ojoj+1) = 4, w(ijij+4) = 5, and w(ojij) = 12, again for j = 0, . . . , 10 and again modulo 11, this basis becomes the unique minimum one, see [9]. 2For the definition of integral cycle bases we refer to [9]. A. Reich: Classification of robust cycle bases and relations to fundamental cycle bases 15 Each edge ijij+4 is contained in three basic circuits, all other edges in exactly two basic circuits. This shows the non-fundamentality of the basis. To see that it is not strictly robust, consider for example the circuit C1,4,12 = C1 + C4 + C12 whose cuts with C1, C4, and C12 do not form a single path in each case. It remains to show that the basis is cyclically robust. This was done by a small program implemented in C++ using LEDA ([10]). The program tested for each of the 212 linear combinations if it constitutes a circuit C, and if so, if there is a circuit Cj 2 supp(C) such that C + Cj is a circuit. This applied to each circuit and thus, the cycle basis is cyclically robust. Example 5.9 (Not weakly fundamental—not cyclically robust). To construct cycle bases of a biconnected graph which are neither robust nor fundamental, the authors in [8] suggest the following operation. Given a 2-connected graph G0 with a non-robust cycle basis B0 and a 2-connected graph G00 with a non-fundamental cycle basis B00, construct a graph G by identifying two arbitrary edges of G0 and G00. The basis B = B0 [ B00 is a basis of G. However, even if B0 and B00 are the minimum cycle bases of G0 and G00, respectively, it is not guaranteed that B is a minimum cycle basis of G. In contrast to this construction, we propose Champetier’s graph with its minimum cycle basis as a representative for a minimum non-robust and non-fundamental cycle basis. Also this graph and the cycle basis are taken from [9]. In his Example 11.7, Liebchen considered Champetier’s graph whose unique minimum cycle basis is integral but neither weakly fundamental nor totally unimodular. In Champetier’s original paper [1], it served as a counter-example of a conjecture expressed in [3]: “If G is null-homotopic (i.e., if every cycle of G is the modulo 2 edge sum of triangles), there is an edge e of G such that G \ e is still null-homotopic.” Champetier’s graph is visualized in Figure 15. AA B BC C D D C1 C2 C3 C4 C5 C6 C7 Figure 15: Champetier’s graph and a certificate for the non-robustness of the minimum cycle basis. Champetier’s graph arises from the embedding by identifying the vertices A, B, C and D with their copies. The cycle basis we deal with is formed by the 36 triangles in the 16 Art Discrete Appl. Math. 1 (2018) #P1.03 embedded version. This basis is minimum since there is neither a further triangle which is not the boundary of a face in the embedding in Figure 15 nor a path of length 3 between two copies of one of the vertices A to D. After the vertex identifications, such a path would also compose to a triangle. Hence, the basic circuits are the only triangles. Since each edge is contained in two triangles at least, the basis is non-fundamental. As a proof for the non-robustness, we take the same certificate as in Example 11.7 in [9], i.e. the circuit C = P7 i=1 Ci, indicated in Figure 15 by two paths. In fact, C + Ci does not form a circuit for i = 1, . . . , 7. This shows that the basis is non-robust. Table 2 summarizes the results of this section. It has been inspired by the Venn dia- gram in [8] which also illustrates the relationship between fundamental and robust cycle bases. In the table, we contrast our results with the results listed there. New examples and improvements are emphasized in italic. Table 2: Overview of the results in this section. strictly fundamental weakly fundamental non-fundamental st ri ct ly ro bu st Kn with K1,n1 Fig. 2 in [8] ? [8] as fund. sp. tree - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - minimum basis basis not minimum as above sunflower graph SF (3) Ex. 5.7, basis only this paperstrictly quasi-robust- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - minimum basis minimum basis basis not minimum cy cl ic al ly ro bu st ? Kainen’s basis of K4 non-fundamental [8] basis of the 4-wheel - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - basis not minimum basis not minimum Wagner’s graph with Wagner’s graph joined Petersen graph P11,4 this papera P7 as fund. sp. tree up with a triangle- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - minimum basis minimum basis minimum basis no n- ro bu st K5 with P4 Vogt’s example merging non-rob. basis [8] as fund. sp. tree with non-fund. basis - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - basis not minimum basis not minimum basis not minimum as above three merged K5 Champetier’s graph this paper - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - basis minimum with minimum basis minimum basis a suitable weighting 6 Concluding remarks In this paper, we considered robust cycle bases and isolated strictly and cyclically robust cycle bases, as well as the newer concepts of quasi-robust and strictly quasi-robust cycle bases from each other. We did this by giving suitable examples. Since each of our cycle bases is the uniquely minimum one of its graph, and hence each type of robust cycle basis A. Reich: Classification of robust cycle bases and relations to fundamental cycle bases 17 comes along with its own minimization problem, we can view the classification of robust cycle bases as completed. A second focus was the continuation of the comparison between robust and funda- mental types of cycle bases. We were able to further fill the Venn diagram of robust and fundamental cycle bases given in [8], where we demanded in addition that the provided cy- cle basis is minimum. Our results were summarized in a table which has only one missing item. We could not present a minimum cycle basis which is non-fundamental and strictly robust, but could provide an example of a cycle basis which is strictly quasi-robust, at least. Despite all, there is still plenty of work to do in the field of robust cycle bases. For example, it is still unknown whether each graph provides a strictly robust cycle basis, or a cycle basis of any other robust type, at least. Furthermore, there is nothing known about the complexity of recognition and construction of robust cycle bases. References [1] C. Champetier, On the null-homotopy of graphs, Discrete Math. 64 (1987), 97–98, doi:10. 1016/0012-365x(87)90244-5. [2] U. Doḡrusöz and M. S. Krishnamoorthy, Enumerating all cycles of a planar graph, Parallel Algorithms Appl. 10 (1996), 21–36, doi:10.1080/10637199608915603. [3] P. Duchet, M. Las Vergnas and H. Meyniel, Connected cutsets of a graph and triangle bases of the cycle space, Discrete Math. 62 (1986), 145–154, doi:10.1016/0012-365x(86)90115-9. [4] P. M. Gleiss, Short Cycles, Ph.D. thesis, Universität Wien, 2001, https://www.tbi. univie.ac.at/papersold/papers/Abstracts/pmg_diss.pdf. [5] D. Hartvigsen and E. Zemel, Is every cycle basis fundamental?, J. Graph Theory 13 (1989), 117–137, doi:10.1002/jgt.3190130115. [6] P. C. Kainen, On robust cycle bases, Electron. Notes Discrete Math. 11 (2002), 430–437, doi: 10.1016/s1571-0653(04)00087-3. [7] P. C. Kainen, Cycle construction and geodesic cycles with application to the hypercube, Ars Math. Contemp. 9 (2015), 27–43, http://amc-journal.eu/index.php/amc/ article/view/450. [8] K. Klemm and P. F. Stadler, A note on fundamental, non-fundamental, and robust cycle bases, Discrete Appl. Math. 157 (2009), 2432–2438, doi:10.1016/j.dam.2008.06.047. [9] C. Liebchen, Periodic Timetable Optimization in Public Transport, Ph.D. thesis, Technische Universität Berlin, 2006. [10] K. Mehlhorn and S. Näher, LEDA: A Platform for Combinatorial and Geometric Computing, Cambridge University Press, Cambridge, 1999, https://people.mpi-inf.mpg.de/ ˜mehlhorn/LEDAbook.html. [11] P.-J. Ostermeier, Quasi-robust Cycle Spaces, 2009, http://www.bioinf. uni-leipzig.de/conference-registration/09herbst/talks/401_ Ostermeier.pdf (accessed on 4 October 2017). [12] P.-J. Ostermeier, M. Hellmuth, K. Klemm, J. Leydold and P. F. Stadler, A note on quasi-robust cycle bases, Ars Math. Contemp. 2 (2009), 231–240, http://amc-journal.eu/index. php/amc/article/view/104. [13] A. Reich, Cycle Bases of Graphs and Spanning Trees with Many Leaves, Ph.D. thesis, Bran- denburgische Technische Universität Cottbus - Senftenberg, 2014, https://opus4.kobv. de/opus4-btu/files/2966/Dissertation_Reich.pdf. 18 Art Discrete Appl. Math. 1 (2018) #P1.03 [14] M. M. Sysło, On cycle bases of a graph, Networks 9 (1979), 123–132, doi:10.1002/net. 3230090203. [15] G. Wünsch, Coordination of Traffic Signals in Networks and Related Graph Theoretical Prob- lems on Spanning Trees, Ph.D. thesis, Technische Universität Berlin, 2008. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.04 https://doi.org/10.26493/2590-9770.1240.60e (Also available at http://adam-journal.eu) Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian Dave Witte Morris ⇤ Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada Received 19 March 2017, accepted 21 December 2017, published online 17 January 2018 Abstract We show that if G is a finite group whose commutator subgroup [G,G] has order 2p, where p is an odd prime, then every connected Cayley graph on G has a hamiltonian cycle. Keywords: Cayley graph, hamiltonian cycle, commutator subgroup. Math. Subj. Class.: 05C25, 05C45 1 Introduction Let G be a finite group. It is easy to show that if G is abelian (and |G| > 2), then every connected Cayley graph on G has a hamiltonian cycle. (See Definition 2.1 for the definition of the term Cayley graph.) To generalize this observation, one can try to prove the same conclusion for groups that are close to being abelian. Since a group is abelian precisely when its commutator subgroup is trivial, it is therefore natural to try to find a hamiltonian cycle when the commutator subgroup of G is close to being trivial. The following theorem, which was proved in a series of papers, is a well-known result along these lines. Theorem 1.1 (Marušič [13], Durnberger [4, 5], 1983–1985). If the commutator subgroup [G,G] of G has prime order, then every connected Cayley graph on G has a hamiltonian cycle. D. Marušič (personal communication) suggested more than thirty years ago that it should be possible to replace the prime with a product pq of two distinct primes: Problem 1.2 (D. Marušič, personal communication, 1985). Show that if the commutator subgroup of G has order pq, where p and q are two distinct primes, then every connected Cayley graph on G has a hamiltonian cycle. ⇤Homepage: http://people.uleth.ca/˜dave.morris/ E-mail address: Dave.Morris@uleth.ca (Dave Witte Morris) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.04 This has recently been accomplished when G is either nilpotent [8] or of odd order [16]. As another step toward the solution of this problem, we establish the special case where q = 2: Theorem 1.3. If the commutator subgroup of G has order 2p, where p is an odd prime, then every connected Cayley graph on G has a hamiltonian cycle. See the bibliography of [12] for references to other results on hamiltonian cycles in Cayley graphs. The proof of Theorem 1.3 is a lengthy case-by-case analysis, based on the choice of certain elements a and b of the Cayley graph’s connection set (see Notation 3.3). Here is an outline of the paper: 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Some known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Assumptions, group theory, and connected sums . . . . . . . . . . . . . . . . 6 4 Case with s = t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5 Cases with |a| > 2 and b /2 hai . . . . . . . . . . . . . . . . . . . . . . . . 12 6 Cases with b 2 hai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 7 Cases with |a| = 2 and #S = 2 . . . . . . . . . . . . . . . . . . . . . . . 18 8 Cases with |a| = 2 and #S = 3 . . . . . . . . . . . . . . . . . . . . . . . 19 9 Cases with |a| = 2 and #S 4 . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Some known results We recall a few results that provide hamiltonian cycles in various Cayley graphs. Definition 2.1 (cf. [9, p. 34]). For any subset S of a finite group G, Cay(G;S) is the graph whose vertex set is G, with an edge joining g to gs, for each g 2 G and s 2 S. This is called the Cayley graph of the connection set S on the group G. Remark 2.2. Unlike most authors (including [9]), we do not require the connection set S to be symmetric in the definition of a Cayley graph; that is, we do not assume S is closed under inverses. This does not change the set of graphs that are considered to be Cayley graphs, because, in our notation, Cay(G;S) = Cay(G;S [ S1), where S1 = {s1 | s 2 S}. Theorem 2.3 ([3, 6, 7, 12]). Every connected Cayley graph on G has a hamiltonian cycle if |G| = kp for some prime p and some k 2 N with 1  k < 32 and k 6= 24. Notation 2.4. • The symbol G always represents a finite group. • For g 2 G and s1, . . . , sn 2 S [ S1, we use [g](s1, . . . , sn) to denote the walk in Cay(G;S) that visits (in order), the vertices g, gs1, gs1s2, gs1s2s3, . . . , gs1s2 · · · sn. We may write (s1, . . . , sn) for [e](s1, . . . , sn). • We use (s1, . . . , sn)k to denote the concatenation of k copies of the sequence (si)ni=1. D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 3 • Appending # to a sequence deletes the last term; that is, (si)ni=1# = (si) n1 i=1 . • If W = [g](s1, . . . , sn) is a walk in Cay(G;S), and h 2 G, we use hW to denote the translate [hg](s1, . . . , sn). • When C is an oriented cycle, we use C to denote the same cycle as C, but with the opposite orientation. • For g, h 2 G: [g, h] = g1h1gh, gh = h1gh, and hg = hgh1 (= gh 1 ). • We use G0 to denote the commutator subgroup [G,G] of G. • For convenience, we let G = G/G0. • For g 2 G, we let g = gG0 be the image of g in G. • We use Z(G) to denote the center of G. Definition 2.5 (cf. [10, §2.1.3, p. 61]). Suppose • N is an abelian, normal subgroup of G, and • C = [Nv](si)ni=1 is an (oriented) cycle in Cay(G/N ;S). The voltage of C is v( Qn i=1 si). This is an element of N , and it may be denoted ⇧C. We have the following straightforward observations: Lemma 2.6. Assume the notation of Definition 2.5. Then: 1. ⇧C is determined by the oriented cycle C: it is independent of the choice of the vertex Nv of C, and of the choice of the representative v of Nv. 2. ⇧ gC = g(⇧C) for all g 2 G. 3. ⇧(C) = (⇧C)1. Definition 2.7. A subset S of G is an irredundant generating set of G if S generates G, but no proper subset of S generates G. Lemma 2.8 (“Factor Group Lemma” [15, §2.2]). Suppose • N is a cyclic, normal subgroup of G, • (si)mi=1 is a hamiltonian cycle in Cay(G/N ;S), and • the voltage ⇧(si)mi=1 generates N . Then (s1, s2, . . . , sm)|N | is a hamiltonian cycle in Cay(G;S). Corollary 2.9 ([12, Cor. 2.11]). Suppose • N is a normal subgroup of G, such that |N | is prime, • the image of S in G/N is an irredundant generating set of G/N , • there is a hamiltonian cycle in Cay(G/N ;S), and • s ⌘ t (mod N) for some s, t 2 S [ S1 with s 6= t. 4 Art Discrete Appl. Math. 1 (2018) #P1.04 Then there is a hamiltonian cycle in Cay(G;S). Lemma 2.10 ([2, Lem. 1 on p. 24]). Let Pk ⇤ P` be the Cartesian product of a path of length k with a path of length `. If k` is even, and k, ` 2, then Pk⇤P` has a hamiltonian path from any corner vertex v to any vertex that is at odd distance from v. Corollary 2.11. Suppose N is a subgroup of an abelian group H , and {x, y} [ S0 is a subset of H that generates H/N . Let k = |hx,Ni : N | and ` = |hx, y,Ni : hx,Ni|. If k` is even, k, ` 2, 0  p < k, 0  q < `, and p+ q is odd, then Cay(H/N ; {x, y}[S0) has a hamiltonian path (si)ri=1, such that s1s2 · · · sr = xpyq . Proof. If we identify the vertices of Pk ⇤ P` with {(i, j) | 0  i < k, 0  j < `} in the natural way, then the map (i, j) 7! xiyj is an isomorphism from Pk ⇤ P` to a subgraph X of Cay hx, yi;x, y . So Lemma 2.10 provides a hamiltonian path (ti)k`1i=1 in X from e to x p y q . So t1t2 · · · tk`1 = xpyq . Let L = (uj)nj=1 be a hamiltonian path in Cay H/hx, y,Ni , and let (si) r i=1 = (L, t2i1, L 1 , t2i) k`/2 i=1 #. From the definition of k and `, we see that the natural map from X to the Cayley graph Cay hx, y,Ni/N ;x, y is an isomorphism onto a spanning subgraph. Therefore, (si)ri=1 is a hamiltonian path in Cay(H/N ;S). Since H is abelian, it is easy to see that s1s2 · · · sr = x p y q . Given a hamiltonian cycle C0 in Cay(G;S), the following result often provides a sec- ond hamiltonian cycle C1, such that the voltage of at least one of these two cycles gener- ates G0. (Then the Factor Group Lemma (2.8) provides a hamiltonian cycle in Cay(G;S).) Lemma 2.12 (cf. Marušič [13] and Durnberger [4], or see [16, Lem. 3.1]). Assume: • N is an abelian normal subgroup of G, such that G/N is abelian, • C0 is an oriented hamiltonian cycle in Cay(G/N ;S), • s, t, u 2 S±1 and h 2 G, • C0 contains: the oriented path [hs1u1](s, t, s1), and either the oriented edge [h](t) or the oriented edge [ht](t1). Then there is a hamiltonian cycle C1 in Cay(G/N ;S), such that ⇣ ⇧C0 1 ⇧C1 ⌘h = ( [u, t1] [s, t1]u if C0 contains [h](t), [t1, u] [s, t1]u if C0 contains [ht](t1). Furthermore, C0 and C1 have exactly the same oriented edges, except for some of the edges in the subgraph induced by {h, hu1, hs1u1, ht, htu1, hts1u1}. Lemma 2.13 ([4, Lem. 2.8]). Assume • S is an irredundant generating set of G, • s, t 2 S, with s 6= t, D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 5 • s commutes with t, • hS r {s}i /G, and • there is a hamiltonian cycle in Cay hS r {s}i;S r {s} . Then there is a hamiltonian cycle in Cay(G;S). We do not need the general theory of nilpotent groups, but we will make use of the following two facts. (The first is essentially the definition of a nilpotent group, which can be found in any graduate-level textbook on group theory.) Lemma 2.14 ([14, (iii) on p. 175 and Prop. VI.1.h on page 176]). 1. Every abelian group is nilpotent. 2. If G/Z(G) is nilpotent, then G is nilpotent. Therefore, if G 0 ✓ Z(G) (in other words, if G/Z(G) is abelian), then G is nilpotent. Theorem 2.15 ([8]). If G is a nontrivial, nilpotent, finite group, and the commutator sub- group of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle. The following observation is well known (and easy to prove). Lemma 2.16 ([12, Lem. 2.27]). Let S generate a finite group G and let s 2 S, such that hsi /G. If • Cay G/hsi;S has a hamiltonian cycle, and • either 1. s 2 Z(G), or 2. Z(G) \ hsi = {e}, or 3. |s| is prime, then Cay(G;S) has a hamiltonian cycle. Corollary 2.17. Suppose • G0 is cyclic of order pq, where p and q are distinct primes, • S is an irredundant generating set of G, and • some nontrivial element s of S is in G0. Then Cay(G;S) has a hamiltonian cycle. Proof. We may assume G0 = Zp⇥Zq . Since every subgroup of a cyclic, normal subgroup is also normal, we know that hsi /G. Also, there are hamiltonian cycles in Cay(G/Zp;S), Cay(G/Zq;S), and Cay(G/G0;S) (by Theorem 1.1 and the elementary fact that Cayley graphs on abelian groups have hamiltonian cycles). Hence, we may assume hsi = G0 and G 0 \Z(G) = Zq (perhaps after interchanging p and q), for otherwise Lemma 2.16 applies. Let bG = G/Zp. We may assume | bG| 6= 27, for otherwise |G| = 27p so Theorem 2.3 applies. Then, since bG is nilpotent (see Lemma 2.14) and its commutator subgroup is Zq , the proof in [11, §4] implies there is a hamiltonian cycle (ti)ni=1 in Cay bG/ bG0;S0) whose 6 Art Discrete Appl. Math. 1 (2018) #P1.04 voltage generates bG0. Then, since Zp \ Z(G) = {e}, the proof of Lemma 2.16(2) in [12, Lem. 2.27(2)] tells us that (ti, sp1)ni=1 is a hamiltonian cycle in Cay G/Zq;S . Note that, since bG is a nilpotent group whose commutator subgroup is in the center and has prime order q, the order of | bG/ bG0| must be a multiple of q; that is, n is a multiple of q (cf. Lemma 3.6 below). Calculating modulo Zp, we have ⇧(ti, s p1)ni=1 ⌘ s(p1)n ⇧(ti)ni=1 (bs 2 bG0 = cZq ✓ Z( bG)) ⌘ ⇧(ti)ni=1 (n is a multiple of q) 6⌘ e (⇧(ti)ni=1 generates bG0). Therefore ⇧(ti, sp1)ni=1 generates Zq . So the Factor Group Lemma (2.8) tells us that (ti, sp1)ni=1 q is a hamiltonian cycle in Cay(G;S). 3 Assumptions, group theory, and connected sums Assumptions 3.1. The remainder of this paper provides a proof of Theorem 1.3, so • p is an odd prime, • G is a finite group whose commutator subgroup has order 2p, and • S is an irredundant generating set of G. We wish to show that the Cayley graph Cay(G;S) has a hamiltonian cycle. 3A Basic group theory Assumption 3.2. Because of Corollary 2.17, we may assume S \G0 = ;. Notation 3.3. The assumption that the commutator subgroup has order 2p implies that G0 is cyclic (cf. [16, §2E, proof of Cor. 1.4]), so we may write G 0 = Z2 ⇥ Zp. From Theorem 2.15, we may assume that G is not nilpotent, so G0 * Z(G) (see Lem- ma 2.14). This implies Zp \ Z(G) = {e}. Hence there exists a 2 S, such that a does not centralize Zp. (3.3A) Then there exists b 2 S, such that Zp ✓ h[a, b]i. (3.3B) The assumptions (3.3A) and (3.3B) are the basis of most of the arguments in the later sections of the paper. For ease of reference, we now collect a few well-known facts from group theory (spe- cialized to our setting). Lemma 3.4. If S0 ✓ G, such that hS0,Z2i = G, then hS0i = G. D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 7 Proof. Since Z2 ✓ Z(G), we have hS0i0 = ⌦ S0, Z(G) ↵0 ◆ hS0,Z2i0 = G0. Therefore hS0i = ⌦ S0, hS0i0 ↵ = hS0, G0i ◆ hS0,Z2i = G. Corollary 3.5. Suppose S0 is a proper subset of S, such that Zp ✓ hS0i. (In particular, this will be the case if {a, b} ✓ S0.) Then hS0i 6= G. Proof. Suppose hS0i = G. This means hS0, G0i = G. Since G0 = Z2⇥Zp and Zp ✓ hS0i, this implies hS0,Z2i = G. So Lemma 3.4 tells us that hS0i = G. This contradicts the fact that the generating set S is irredundant. Lemma 3.6. Let H be a group. If x, y, z 2 H , and y centralizes H 0, then [xy, z] = [x, z] [y, z]. Therefore [yk, z] = [y, z]k for all k 2 Z. Corollary 3.7. If x, y 2 G, such that y centralizes G0, and Zp ✓ h[x, y]i, then |y| is divisible by p. Corollary 3.8. Let S0 ✓ G, such that Z2 * hS0i0. If g 2 G, such that Z2 ✓ hg, S0i0, then |hg, S0i : hS0i| is even. In particular, if Z2 ✓ h[g, h]i, then, by taking S0 = {h}, we see that |hg, hi : hhi| is even, so |g| is even (and, similarly, |h| must also be even). Corollary 3.9. |G| is divisible by 4. 3B Connected sums Definition 3.10 ([8, Defn. 5.1]). Assume C1 and C2 are two vertex-disjoint oriented cycles in Cay(G;S), and let g 2 G, and s, t 2 S [ S1. If • C1 contains the oriented edge [g](t), and • C2 contains the oriented edge [gst](t1), then we use C1 #st C2 to denote the oriented cycle obtained from C1 [ C2 by • removing the oriented edges [g](t) and [gst](t1), and • inserting the oriented edges [g](s) and [gst](s1). This is called the connected sum of C1 and C2. If [g](t) is any oriented edge of an oriented cycle C, and s 2 S, such that sC is vertex disjoint from C, then we can form the connected sum C#st sC. This construction can be iterated: Definition 3.11. Suppose • [g1](t1), . . . , [gk](tk) are oriented edges of an oriented cycle C in Cay(G;S), such that gi 6= gi+1 for all i, and 8 Art Discrete Appl. Math. 1 (2018) #P1.04 • s1, s2, . . . , sk 2 S [ S1, such that the cycles C, s1C, s2s1C, . . . , sksk1 · · · s1C are pairwise vertex-disjoint. Then we can form the connected sum C #s1t1 s1C # s2 t2 s2s1C # s3 t3 · · · # sk tk ±sksk1 · · · s1C. We call this a connected sum of signed translates of C. Lemma 3.12 (cf. [8, Lem. 5.2]). If C1, C2, g, s, and t are as in Definition 3.10, then ⇧(C1 # s t C2) = ⇧C1 · g[s1, t1] ·⇧C2. Proof. We may assume g = t1 (or, in other words, gt = e), after translating the cycles by (gt)1 (cf. Lemma 2.6(2)). Write C1 = (si)mi=1 and C2 = [st1](tj)nj=1, so (C1 # s t C2) = (si) m1 i=1 , s, (tj) n1 j=1 , s 1 . By assumption, C1 contains the edge t1 ! e and C2 contains the edge s ! st1, so sm = t and tn = t1. Therefore ⇧(C1 # s t C2) = m1Y i=1 (si) · s · n1Y j=1 (tj) · s1 = mY i=1 (si) · t1s · nY j=1 (tj) · ts1 = ⇧C1 · t1s · (⇧C2)st 1 · ts1 = ⇧C1 · t1sts1 ·⇧C2 = ⇧C1 · t 1 [s1, t1] ·⇧C2 = ⇧C1 · g[s1, t1] ·⇧C2. Corollary 3.13. Assume that C1, C2, g, s, and t are as in Definition 3.10. If C0 is another oriented cycle that is vertex-disjoint from C2 and contains the oriented edge g(t), then ⇧(C0 # s t C2) ⇧(C1 # s t C2) 1 = (⇧C0)(⇧C1) 1 . Corollary 3.14 ([8, Lem. 5.2]). If C1, C2, g, s, and t are as in Definition 3.10, then ⇧(C1 # s t C2) ⌘ ⇧C1 ·⇧C2 · [s, t] (mod Zp). The following result describes a fairly common situation in which the connected sum provides hamiltonian cycles in Cay(G;S): Lemma 3.15. Let S0 be a nonempty subset of S, g 2 G, c 2 S r S0, and s, t 2 S r {c}. Assume C0 and C1 are oriented hamiltonian cycles in Cay hS0i;S0 , such that • (⇧C0)1(⇧C1) is a nontrivial element of Zp, • C0 and C1 both contain the oriented edge [g](s), D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 9 • for every x 2 S0, C0 contains at least two edges that are labelled either x or x1, • Z2 ✓ h[c, t]i, and • either |G : hS0i| > 2 or s = t. If either 1. there exists u 2 S r {c}, such that Z2 * h[u, c]i, or 2. |G : hS0, ti| is even, then there is a hamiltonian cycle C in Cay(G;S), such that h⇧Ci = G0, so the Factor Group Lemma (2.8) yields a hamiltonian cycle in Cay(G;S). Proof. Let r = |G : hS0i|. We have Zp ✓ h(⇧C0)1(⇧C1)i ✓ hS0i, so Corollary 3.5 implies r 6= 1. Suppose r = 2. By assumption, this implies s = t, which means that C0 and C1 both contain the oriented edge [g](t). Then the translate cC0 contains the oriented edge [gc](t). The connected sums C = C0 #ct cC0 and C 0 = C1 #ct cC0 are hamiltonian cycles in Cay(G;S). From Corollary 3.14, we have ⇧C ⌘ ⇧C0 ·⇧C0 · [c, t] ⌘ [c, t] 6⌘ 0 (mod Zp), so ⇧C projects nontrivially to Z2. Corollary 3.13 says (⇧C)(⇧C 0)1 = (⇧C0)(⇧C1)1, which generates Zp (because it is conjugate to the inverse of (⇧C0)1(⇧C1), which is assumed to be a nontrivial element of Zp). Therefore, we see that either ⇧C or ⇧C 0 generates G0, as desired. So we may assume henceforth that r > 2. We now show that we may assume t 2 S0. To this end, suppose it is not the case that t 2 S0. Let n = |hS0, ti : hS0i|. Then, by choosing a sequence {[gi](si)}n1i=1 of oriented edges of C0, we can form a connected sum C 00 of signed translates of C0: C 0 0 = C0 # t s1 tC0 # t s2 · · ·# t sn1 ±t n1 C0. This is a hamiltonian cycle in Cay hS0, ti;S0 [ {t} . We may assume s1 = s. Then another hamiltonian cycle C 01 can be constructed by replacing the leftmost occurrence of C0 with C1, and Lemma 3.12 tells us that (⇧C 00)(⇧C 01)1 = (⇧C0)(⇧C1)1, which is a nontrivial element of Zp (and (⇧C0)1(⇧C1) is conjugate to the inverse of this). From the definition of connected sum, it is obvious that C 00 contains at least two edges labelled t ±1. So the hamiltonian cycles C 00 and C 01 satisfy the hypotheses of the lemma with S0[{t} in the role of S0 and with t in the role of s. Case 1. Assume there exists u 2 S r {c}, such that Z2 * h[u, c]i. Subcase 1.1. Assume u 2 S0. Fix a hamiltonian path (si)ni=1 in Cay(G/hS0i;SrS0) with s1 = c, and let ⇡i = Qi j=1 sj . Any connected sum C0# s1 t1 (⇡1C0)# s2 t2 · · ·# sn tn (±⇡nC0) is a hamiltonian cycle C in Cay(G;S). Since [t, c] and [u, c] do not have the same projection to Z2, the voltages of C0 #ct ⇡1C0 and C0 #cu ⇡1C0 do not have the same projection to Z2. Therefore, by choosing t1 to be the appropriate element of {t, u}, we may assume the projection of ⇧C to Z2 is nontrivial (see Corollary 3.14). Note also that if |G : hS0i| = 2, then we must have t1 = t. 10 Art Discrete Appl. Math. 1 (2018) #P1.04 We may assume that tn = s, and that the connected sum (1)n1⇡n1C0#sns (1)n⇡nC0 is relative to the oriented edge [⇡ng](s) of ⇡nC0 that is also in ⇡nC1. There- fore, another hamiltonian cycle C 0 can be constructed by replacing ⇡nC0 with ⇡nC1 in the connected sum. Then Lemma 3.12 (together with Lemma 2.6(2)) implies that (⇧C)1(⇧C 0) is conjugate to (⇧C0)1(⇧C1), which is a generator of Zp. Therefore, either ⇧C or ⇧C 0 generates G0, as desired. Subcase 1.2. Assume u /2 S0. Let Su = {u}[S0, let n = |hSui : hS0i| 1, let (si)mi=1 be a hamiltonian path in Cay G/hSui;S r Su with s1 = c, and let ⇡i = Qi j=1 sj . (Since S r S0 is an irredundant generating set for G/hS0i, we have m,n 1.) Any connected sum Cu = C0 # u t1 uC0 # u t2 · · ·# u tn ±u n C0 is a hamiltonian cycle in Cay hSui;Su , so any connected sum C = Cu # s1 t01 ⇡1Cu #s2t02 · · ·# sm t0m ±⇡mCu is a hamiltonian cycle in Cay(G;S). Since t 2 S0, we know that C0 contains more than one edge labeled t±1, so uC0 has an edge labeled t±1 that was not removed in the construction of the connected sum C0 #ut1 ⇡1C0. Furthermore, the definition of the connected sum implies that C0 #ut1 ⇡1C0 also contains an edge labeled u. Therefore, we may form connected sums Cu # c t±1 ⇡1Cu and Cu # c u ⇡1Cu without removing any of the edges of Cu. Since [c, t] and [c, u] do not have the same projection to Z2, the voltages of these two connected sums do not have the same projection to Z2 (see Corollary 3.14). Therefore, by choosing t01 to be the appropriate element of {t±1, u}, we may assume the projection of ⇧C to Z2 is nontrivial. We have C = Cu # s1 t01 ⇡1Cu #s2t02 · · ·# sm1 t0m1 ±⇡m1Cu#smt0m ±⇡mC0 #ut1 ±⇡muC0 # u t2 · · ·# u tn ±⇡mu n C0 , so the proof can be completed almost exactly as in the final paragraph of Subcase 1.1 (by constructing another connected sum in which ⇡munC0 is replaced with ⇡munC1). Case 2. Assume [u, c] projects nontrivially to Z2, for every u 2 S r {c}. In particular, [d, c] projects nontrivially to Z2, for every d 2 S r S0 [ {c} . Since we may assume that Case 1 does not apply with d in the place of c, we conclude that we may assume [u, d] projects nontrivially to Z2, for all d 2 S r S0 and u 2 S r {d}. (3.15A) Choose a hamiltonian path (si)ni=1 in Cay(G/hS0i;S r S0). Any connected sum C = C0 # s1 t1 ⇡1C0 # s2 t2 · · ·# sn tn ±⇡nC0 D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 11 is a hamiltonian cycle in Cay(G;S). Calculating modulo Zp, and letting z be the nontrivial element of Z2, we have ⇧C ⌘ ⇧C0 · [s1, t1] ·⇧(⇡1C0) · · · [sn, tn] ·⇧(±⇡nC0) (Corollary 3.14) ⌘ ⇧C0 · z ·⇧C0 · · · z ·⇧C0 (Lemma 2.6(2) & (3.15A)) = (⇧C0) n+1 · zn ⌘ z (n is odd). The proof is now completed exactly as in the final paragraph of Subcase 1.1. Corollary 3.16. Let S0 ✓ S, g 2 G, and s 2 S0. Assume C0 and C1 are oriented hamiltonian cycles in Cay hS0i;S0 , such that • (⇧C0)1(⇧C1) is a nontrivial element of Zp, • C0 and C1 both contain the oriented edge [g](s), • for every x 2 S0, C0 contains at least two edges that are labelled either x or x1, and • Z2 * hS0i0. Then there is a hamiltonian cycle C in Cay(G;S), such that h⇧Ci = G0, so the Factor Group Lemma (2.8) yields a hamiltonian cycle in Cay(G;S). Proof. We may assume [c, t] 2 Zp, for all c 2 S and t 2 S0. (Otherwise, we see from Corollary 3.8 that Lemma 3.15(2) applies.) Choose c, d 2 S, such that [c, d] /2 Zp, let S + 0 = S0 [ {d}, and let r = |hS + 0 i : hS0i|. Any connected sum of the following form is a hamiltonian cycle in Cay hS+0 i;S + 0 : C = C0 # d s1 dC0 # d s2 · · ·# d sr1 ±d r1 C0. We may assume s1 = s, and that the connected sum C0 #ds1 dC0 is formed by using the oriented edge [g](s) that is also in C1. Therefore, a second hamiltonian cycle C 0 can be constructed by replacing the leftmost occurrence of C0 with C1. Then Corollary 3.8 implies that Lemma 3.15(2) applies (with S+0 , d, d, C, and C 0 in the roles of S0, s, t, C0, and C1, respectively). 4 Case with s = t Case 4.1. Assume there exist s, t 2 S [ S1 with s = t and s 6= t. Proof. Write t = s with 2 G0. We may assume hi = G0, for otherwise || is prime, so Corollary 2.9 applies with N = hi. Note that the irredundance of S implies hSr{s}i and hS r {t}i do not contain Zp. This implies that every element of S r {s, t} centralizes Zp. So s and t do not centralize Zp. Let m = |t| and n = |G|/m. Subcase 4.1.1. Assume |t| > 2. Since G is abelian, it is easy to find a hamiltonian cycle C = (ti)mni=1 in Cay G;S r {s} , such that t1 = t2 = · · · = tm1 = t. Since h⇧Ci ✓ hS r {s}i, and Zp * hS r {s}i, we must have ⇧C 2 Z2. For each subset I of {1, . . . ,m 1}, we define CI to be the hamiltonian cycle con- structed from C by changing ti to s for all i 2 I . The proof is completed by noting that I may be chosen such that ⇧CI generates G0, so the Factor Group Lemma (2.8) applies: 12 Art Discrete Appl. Math. 1 (2018) #P1.04 • If ⇧C = e, let I = {1}. • If ⇧C is the nontrivial element of Z2, and t does not invert Zp, then we may let I = {1, 2}. • If ⇧C is the nontrivial element of Z2, and t inverts Zp, then |t| is even, so we must have |t| 4. We may let I = {1, 3}. Subcase 4.1.2. Assume |t| = 2. (Since t does not centralize Zp, this implies that t in- verts Zp.) Choose a hamiltonian cycle (si)ni=1 in Cay G/hti;S r {s, t} , and let C0 = (t, si) n i=1 = (tj) 2n j=1. Since n = |G|/2 is even (see Corollary 3.9) and S r {s} is an irredundant generating set of G, it is easy to see that C0 is a hamiltonian cycle in Cay G;S r {s} . Note that ti = t whenever i is odd, and that ⇧C0 2 Z2 (because Zp * hS r {s}i). We may assume n 6 (for otherwise |G| = 4np  20p, so Theorem 2.3 applies). We construct a hamiltonian cycle C1 from C0: • If ⇧C0 = e, construct C1 by changing t1 to s. • If ⇧C0 6= e, construct C1 by changing both t1 and t5 to s. In each case, ⇧C1 generates G0. (To see this in the second case, note that t2t3t4t5 = s1ts2t centralizes G0, because t inverts G0, and each si centralizes G0.) Therefore, the Factor Group Lemma (2.8) applies. 5 Cases with |a| > 2 and b /2 hai Recall that the elements a and b of S satisfy (3.3A) and (3.3B). Case 5.1. Assume |a| > 2, b /2 hai, and there exists c 2 S, such that Z2 ✓ h[a, c]i. (It may be the case that b = c.) Proof. Let m = |a| and n = |G : hai|. Since b, c /2 hai (and G/hai is abelian), it is easy to find a hamiltonian cycle (si)ni=1 in Cay G/hai;S r {a} , such that sn 2 {c±1}, and sk = b for some k < n. Since Z2 ✓ h[a, c]i, we know m and n are both even (see Corollary 3.8). Since n is even, we have the following (well-known) hamiltonian cycle C0 in Cay(G;S): C0 = a, (am2, s2i1, a (m2) , s2i) n/2 i=1#, a 1 , (s1nj) n1 j=1 . (5.1A) Letting bG = G/Zp, we have bG0 = Z2, so bam2 2 Z( bG ) (because m is even). There- fore a m2 s2i1a (m2) ⌘ s2i1 (mod Zp), so, calculating modulo Zp, we have ⇧C0 ⌘ a · n1Y i=1 sj ! · a1 · n1Y i=1 sj !1 ⌘ a · s1n · a1 · sn = [a1, sn] = [a1, c±1], which is nontrivial (mod Zp). D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 13 Recall that sk = b. Let g = Qk1 i=1 si and = (1)k+1. Then C0 contains both the oriented edge [gb](b1) and the oriented path [ga2](a, b, a). So Lemma 2.12 (with s = a , t = b, u = a and h = g) provides a hamiltonian cycle C1, such that (⇧C0)1(⇧C1) is conjugate to [b1, a][a, b1]a . Since a centralizes Z2, but not Zp, this voltage is a generator of Zp. Thus, either ⇧C0 or ⇧C1 generates Z2 ⇥ Zp = G0, so the Factor Group Lemma (2.8) provides a hamiltonian cycle in Cay(G;S). Case 5.2. Assume |a| > 2, b /2 hai, and there does not exist c 2 S, such that Z2 ✓ h[a, c]i. Proof. Choose c, d 2 S with Z2 ✓ h[c, d]i. Let m = |a|, n = |hS r {d}i|/m, and r = |G|/(mn). By assumption, we know a /2 {c, d}. Also, we may assume d 6= b (after interchanging c and d if necessary). Then Corollary 3.5 tells us r > 1. Furthermore, we see from Corollary 3.8 that the image of c in G/hai has even order, so n is even. Subcase 5.2.1. Assume n > 2. It is not difficult to construct a hamiltonian cycle (si)ni=1 in Cay hS r {d}i/hai;S r {a, d} , such that s1 = b and sk = c±1 for some k /2 {1, n}. Then, since n is even, we may define C0 as in (5.1A), so C0 is a hamiltonian cycle in Cay hS r {d}i;S r {d} . Let g = s1s2 · · · sk, and note that C0 contains the oriented edges [e](a) and [g](c⌥1). Since Z2 ✓ h[c, d]i, but Z2 * h[a, d]i, we see from Lemma 3.12 that there is a connected sum C = C0 # d t1 dC0 # d t2 · · ·# d tr1 ±d r1 C0, with t1 2 {a, c±1}, such that Z2 ✓ h⇧Ci. Note that C is a hamiltonian cycle in Cay(G;S). The cycle C0 contains both [b](b1) and [a2](a, b, a1), and neither of these paths contains either the edge [e](a) or the edge [g](c⌥1). Therefore, C also contains both of these paths, so Lemma 2.12 (with s = a, t = b, u = a, and h = e) provides a hamilto- nian cycle C 0 in Cay(G;S), such that ⇧C 1 ⇧C 0 is a conjugate of [b1, a] [a, b1]a, which is a generator of Zp (since a centralizes Z2, but not Zp). Then either ⇧C or ⇧C 0 generates G0, so the Factor Group Lemma (2.8) applies. Subcase 5.2.2. Assume n = 2 and r > 2. Since n = 2 (and b /2 hai), we have ha, b, di = G, so Corollary 3.5 implies S = {a, b, d}. (Therefore b = c, which means Z2 ✓ h[b, d]i.) We have the following hamiltonian cycle in Cay ha, bi; a, b : C0 = [e](a m1 , b, a (m1) , b 1). Using the oriented edge [e](a), we can form the connected sum C0 #da dC0. Then, since dC0 contains both [db](b1) and [dab](a1), we can extend this to a connected sum C = C0 # d a dC0 #dt2 · · ·# d tr1 ±d r1 C0, with t2 2 {a, b}, such that Z2 ✓ h⇧Ci (see Corollary 3.14). Since C contains both [b](b1) and [a2](a, b, a1), we may argue as in the last paragraph of Subcase 5.2.1. Namely, Lemma 2.12 (with s = a, t = b, u = a, and h = e) provides a hamiltonian cy- cle C 0 in Cay(G;S), such that ⇧C 1 ⇧C 0 is a conjugate of [b1, a] [a, b1]a, which is 14 Art Discrete Appl. Math. 1 (2018) #P1.04 a generator of Zp. Then either ⇧C or ⇧C 0 generates G0, so the Factor Group Lemma (2.8) applies. Subcase 5.2.3. Assume n = r = 2. As in Subcase 5.2.2, we must have S = {a, b, d} and b = c (so Z2 ✓ h[b, d]i). Subsubcase 5.2.3.1. Assume m 6= 3. Since m = |a| > 2 (by an assumption of this case), we have m 4. We have the following hamiltonian cycle in Cay(G;S): C0 = d, b, a, b 1 , d 1 , a m2 , d, a (m3) , b, a m3 , d 1 , a (m1) , b 1). Since a is central in G/Zp (by an assumption of this case), we know that ⇧C0 ⌘ dbb1d1dbd1b1 = dbd1b1 = [d1, b1] ⌘ [d, b] = [d, c] (mod Zp), so Z2 ✓ h⇧C0i. Note that C0 contains both [dab](b1) and [da3](a1, b, a) (because m 4), so apply- ing Lemma 2.12 (with s = a1, t = b, u = a1 and h = da) yields a hamiltonian cycle C1 in Cay(G;S), such that ⇧C0 1 ⇧C1 is a conjugate of [b1, a1] [a1, b1]a 1 , which is a generator of Zp. Then either ⇧C or ⇧C 0 generates G0, so the Factor Group Lemma (2.8) applies. Subsubcase 5.2.3.2. Assume m = 3 and d does not centralize G0. Since the walk (a2, b1, a2) is a hamiltonian path in Cay ha, bi; a, b , we have the following hamiltonian cycle in Cay(G;S): C = (a2, b1, a2, d1, a2, b, a2, d). Note that ⇧C = (a2b1a2) d1(a2ba2) d = (ba 2 )1 d1(ba 2 ) d = [ba 2 , d]. Since a2 does not invert G0, we know that ba 2 6⌘ ba2 (mod Z2). Therefore, since d does not centralize G0, we may assume [ba 2 , d] 6⌘ e (mod Z2) (by replacing a with its inverse if necessary). Also, since G0 is central modulo Zp, we have [ba 2 , d] ⌘ [b, d] 6⌘ e (mod Zp). Therefore, ⇧C generates G0, so the Factor Group Lemma (2.8) applies. Subsubcase 5.2.3.3. Assume m = 3 and d centralizes G0. Suppose [b, d] 2 Z2. Let bG = G/Z2 and bH = hba,bbi. From Theorem 1.1, we know there is a hamiltonian cycle in Cay( bH; a, b . Deleting an edge labeled b±1 (and passing to the reverse and/or a translate if necessary) yields a hamiltonian path L = (ti)2mp1i=1 in Cay( bH; a, b from be to bb. Let C = (L1, d1, L, d). Then ⇧C = ⇥Y2mp1 i=1 ti, d] 2 [bZ2, d] = {[b, d]}, because Z2 is in the center of G. Since [b, d] 2 Z2, this calculation implies that C is a closed walk in G/Z2 = bG. So C is a hamiltonian cycle in Cay( bG;S). The calculation also implies that the Factor Group Lemma (2.8) applies, because h[b, d]i = Z2. We may now assume [b, d] /2 Z2. Therefore, since d centralizes G0, and p2 - 12 = |G|, we see from Lemma 3.6 that b does not centralize G0. Also, we may assume [a, d] 6= e, for otherwise Lemma 2.13 applies with s = d and t = a. However, we know Z2 * h[a, d]i (by an assumption of this case). Therefore h[a, d]i = Zp. So Subsubcase 5.2.3.2 applies after interchanging b and d. D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 15 6 Cases with b 2 hai Case 6.1. Assume b 2 hai and a does not invert G0. Proof. Let m = |a|. We may assume (perhaps after replacing b with its inverse) that we may write b = ak with 1  k  m/2 and 2 G0. Assume k 2, for otherwise Case 4.1 applies. This implies m 1 k + 1 (since m = |a| 2k k + 2). Subcase 6.1.1. Assume there exists c 2 S, such that Z2 ✓ h[a, c]i. Let n = |G : hai|. Note that Corollary 3.8 implies m and n are even, and c /2 hai (so c 6= b). Choose a hamiltonian cycle (si)ni=1 in Cay G/hai;S r {a, b} , such that sn = c, and define C0 as in (5.1A). Then h⇧C0i contains Z2 by the same calculation as in Case 5.1. Since m 1 k + 1, we may construct a hamiltonian cycle C1 in Cay(G;S) by replacing the path (ak+1) at the start of C0 with (b, a(k1), b). Then ⇧C1 ⇧C0 1 = ba(k1)ba(k+1) = (ak)a(k1)(ak)a(k+1) = a k+1 a a (k+1) . This is a generator of Zp, since a inverts Z2, but not Zp. Hence, either ⇧C0 or ⇧C1 generates G0, so the Factor Group Lemma (2.8) provides a hamiltonian cycle in Cay(G;S). Subcase 6.1.2. Assume there does not exist c 2 S, such that Z2 ✓ h[a, c]i. Choose c, d 2 S, such that Z2 ✓ h[c, d]i. (It is possible that b 2 {c, d}, but we know, by the assumption of this subcase, that a /2 {c, d}.) Let n = |ha, di : hai| and r = |G|/(mn). From Corollary 3.8 (and the assumption of this subcase), we know n and r are even. We have the following hamiltonian cycle in Cay ha, di; a, d : C0 = (a, (am2, d, a(m2), d)n/2#, a1, d(n1) . As in the final paragraph of Subcase 6.1.1, another hamiltonian cycle C1 can be constructed by replacing the path (ak+1) at the start of C0 with (b, a(k1), b), and the calculation in Subcase 6.1.1 shows that (⇧C1)(⇧C0)1 generates Zp. Therefore, since [c, d] /2 Zp, but [c, a] 2 Zp, we see that Lemma 3.15(1) applies (with S0 = {a, b, d}, g = a1, s = t = d, and u = a). Case 6.2. Assume b 2 hai and a inverts G0. Proof. As in Case 6.1, we let m = |a| and write b = ak with 2  k  m/2 and 2 G0. We now consider the same five subcases as in [4, pp. 60–62]. Subcase 6.2.1. Assume 2 < k < m/2 and k is even. Let C1 = (am). The proof in the last paragraph of [4, p. 60] provides a hamiltonian cycle C0 = b, a (k4) , b, a m2k2 , b, a 1 , b, a 2 , b 2 , a k3 in Cay hai; a, b , such that (⇧C0)1(⇧C1) is a generator of Zp. Therefore, Corol- lary 3.16 applies (with S0 = {a, b}), because C0 and C1 both contain the oriented edge [a1](a). Subcase 6.2.2. Assume 2 < k < m/2 and k is odd. Let C0 = (b, a, b1, a)(k1)/2, b, am2k+1 16 Art Discrete Appl. Math. 1 (2018) #P1.04 and C1 = (b, a1, b1, a1)(k1)/2, b2, am2k1, b . Calculations in [4, p. 61] show that (⇧C0)1(⇧C1) is a generator of Zp. Therefore, Corollary 3.16 applies (with S0 = {a, b}), because C0 and C1 both contain the oriented edge [e](b). Subcase 6.2.3. Assume k = m/2 and k is even. We follow the argument of [11, Subcase iii, p. 97]. Since k is even, we know ak centralizes G0, so b 2 = (ak)2 = a2k2 = am2 2 Z2 · 2 63 e. Therefore Corollary 2.9 applies (with s = b and t = b1). Subcase 6.2.4. Assume k = m/2 and k is odd. Choose c 2 S so that Z2 ✓ h[a, c]i, if such c exists. Otherwise, choose c so that there exists d 2 S, such that Z2 ✓ h[c, d]i. In either case, Corollary 3.8 implies c 2 S r {a, b}, and |ha, ci : hai| is even. We may assume b2 = e, for otherwise Corollary 2.9 applies (with s = b and t = b1). Therefore, noting that ak inverts G0 (since k is odd), we have e = b2 = (ak)(ak) = a2k · 1 = am. Subsubcase 6.2.4.1. Assume |G : hai| > 2. It suffices to find a hamiltonian cycle C⇤ in Cay(G;S), such that ⇧C⇤ projects nontrivially to Z2, and C⇤ contains the paths [ak3](a, b, a1) and [ak1b](b1). For then Lemma 2.12 (with s = a, t = b, u = a, and h = ak1) provides a hamiltonian cycle C 0⇤, such that h(⇧C⇤)1(⇧C 0⇤)i = Zp. There- fore, either ⇧C⇤ or ⇧C 0⇤ generates G0, so the Factor Group Lemma (2.8) applies. Note that C = (ak2, b, a(k2), c, ak1, c1, b1, c, a(k1), c1) is a cycle through the vertices of Cay(G; {a, b, c}) in hai [ chai. A connected sum of translates of C yields a hamiltonian cycle C0 in Cay(G;S). If Z2 * h[a, c]i, then the connected sum defining C0 can be chosen so that Z2 ✓ h⇧C0i (see the proof of Lemma 3.15). So we may let C⇤ = C0. We may now assume Z2 ✓ h[a, c]i. Construct a hamiltonian cycle C1 in Cay(G;S) by replacing the rightmost translate of C in the connected sum with C 0 = (ak1, b, a(k1), c, ak1, b1, a(k1), c1). A straightforward calculation shows that (⇧C)1(⇧C 0) /2 Zp, so we have Z2 ✓ h⇧Cii for some i 2 {0, 1}. Let C⇤ = Ci. Assumptions 6.2.4.2. We may now assume |G : hai| = 2, so the irredundance of S implies S = {a, b, c}. Since b 2 hai, the irredundance of S also implies h[a, c]i = Z2. Furthermore, we may also assume that c either centralizes G0 or inverts G0. (Otherwise, a preceding case applies after interchanging a with c.) Subsubcase 6.2.4.3. Assume c inverts G 0 . Let L = ( (a, b)k# if p | k (b, a)k# if p - k and C = (L1, c1, L, c). D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 17 Then L is a hamiltonian path in Cay hai; a, b , so C is a hamiltonian cycle in Cay(G;S). Since (ab)k = k, we have ⇧L = ( k b 1 = k1ak if p | k, k a 1 if p - k. Thus, in either case, we have ⇧L = yaz , where p - y and z is odd, so ⇧C = (⇧L)1c1(⇧L)c = [⇧L, c] = [yaz, c] = [y, c]a z · [az, c] = (2y)a z · [a, c]z = 2y · [a, c]. This generates G0, so the Factor Group Lemma (2.8) applies. Subsubcase 6.2.4.4. Assume c centralizes G 0 and k 5. Let C0 = (L, c, L 1 , c 1), where L = (b, a)k#. Since C0 contains both [e](b, a, b) and [abc](a1), and also contains both [a2](b, a, b) and [a3bc](a1) we can apply Lemma 2.12 twice (first with s = b, t = a, u = c, and h = bc, and then with s = b, t = a, u = c, and h = a2bc), to obtain a hamiltonian cycle C2, such that (⇧C0) 1(⇧C2) = [a 1 , b]2, which generates Zp. Then, since ⇧C0 = (ba)ka1 c (ba)ka1 1 c 1 = [a, c] is a generator of Z2, we conclude that ⇧C2 generates G0, so the Factor Group Lemma (2.8) applies. Subsubcase 6.2.4.5. Assume c centralizes G 0 and k = 3. Assume, for the moment, that /2 Zp. Let C = (c, b, c1, a, b1, c, b, a, b1, c1, b, a). Then C is a hamiltonian cycle in Cay(G;S), and a straightforward calculation shows that ⇧C = ba3 = 1 generates G0, so the Factor Group Lemma (2.8) applies. Now, suppose that p 5, and, because of the preceding paragraph, that 2 Zp. Let C = (b, a, b1, a, b, c, a5, c1). Then C is a hamiltonian cycle in Cay(G;S) and ⇧C = bab1abcac1 = bab1aba[a, c1] = 3[a, c]. Therefore h⇧Ci = G0 (since p 6= 3 and projects trivially to Z2), so the Factor Group Lemma (2.8) applies. We may now assume p = 3 (so |G| = 72), and that 2 Zp. Let bG = G/Zp. We have the following hamiltonian cycle in Cay( bG;S): C = (a2, c, a5, c1, a2, b, a2, c, a5, c1, a2, b). 18 Art Discrete Appl. Math. 1 (2018) #P1.04 Calculating modulo Z2 (so c is in the center), we have ⇧C = a2ca5c1a2ba2ca5c1a2b ⌘ a2a5a2ba2a5a2b = a1bab = [a, b] = 2. This is nontrivial (mod Z2), so ⇧C must be nontrivial. Therefore ⇧C generates Zp, so the Factor Group Lemma (2.8) applies. Subcase 6.2.5. Assume k = 2 < m/2. Subsubcase 6.2.5.1. Assume |G : hai| > 2. Note that C = b, a, b 1 , c, b, a 1 , b, c 1 , (a, c, a, c1)(m4)/2 is a cycle through the vertices of Cay(G; {a, b, c}) in hai [ chai. A connected sum of translates of C yields a hamiltonian cycle C0 in Cay(G;S). Since k is even, we know that Z2 * h[b, c]i, so it is easy to choose the connected sum in such a way that Z2 ✓ h⇧C0i (see the proof of Lemma 3.15). The cycle C contains the paths [e](b, a, b1) and [b 2 ](a). By taking just a bit of care in the creation of C0 (namely, not using any of these edges for the first connected sum), we may assume that C0 also contains these paths. Then Lemma 2.12 (with s = b, t = a, u = b, and h = b2) provides a hamiltonian cycle C1, such that (⇧C0)1(⇧C1) = [a, b]2 (because b centralizes G0). This is a generator of Zp, so either ⇧C0 or ⇧C1 generates G0. Therefore, the Factor Group Lemma (2.8) applies. Subsubcase 6.2.5.2. Assume |G : hai| = 2. The irredundance of S implies that S = {a, b, c} (see Corollary 3.5). We have the following hamiltonian cycle in Cay(G;S): C = (b2, am5, c, a(m4), c1, b1, c, a, b1, c1). Since b 2 hai, the irredundance of S implies h[a, c]i = Z2. So m is even (see Corol- lary 3.8). However, Z2 * h[b, c]i, because k = 2 is even. So ⇧C = b2(am5ca(m4)c1)(b1cab1c1) ⌘ b2(a1)(b2a[a, c]) ⌘ [a, c] (mod Zp), which generates Z2. We may also assume that c either centralizes G0 or inverts G0 (for otherwise a preceding case applies after interchanging a with c). Therefore ⇧C = b2(am5ca(m4)c1)(b1cab1c1) ⌘ a42(a1)(1a2ca1a2c1) = 3 · (1)c = 3 · ±1 2 {2, 4} (mod Z2), which generates Zp. We now know that ⇧C projects nontrivially to both Z2 and Zp, so it generates G0. Therefore, the Factor Group Lemma (2.8) applies. 7 Cases with |a| = 2 and #S = 2 Assumption 7.1. In this section, we assume • |a| = 2, for all a 2 S, such that a does not centralize G0, and • #S = 2. D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 19 We may assume |a| = 2, for otherwise Case 4.1 applies with s = a and t = a1. We may also assume that b centralizes G0, for otherwise we must have |b| = 2, so |G| = 8p, so Theorem 2.3 applies. Since a does not centralize G0, this implies a /2 hbi. Let n = |G : hai| = |G|/2 = |b|. Case 7.2. Assume n 6⌘ 1 (mod p). Proof. Let C = (a1, b(n1), a, bn1), so C is a hamiltonian cycle in Cay(G;S) with ⇧C = [a, bn1] = [a, b]n1, since b centralizes G0. Note that n is even (see Corollary 3.8), and, by assumption, n 6⌘ 1 (mod p). Therefore, n 1 is relatively prime to 2p, so ⇧C generates G0, so the Factor Group Lemma (2.8) applies. Case 7.3. Assume n ⌘ 1 (mod p). Proof. We claim that Zp ✓ hbi. Suppose not. Then |hb,Z2i| = 2n. Since gcd(2n, p) = 1, the abelian group hb,G0i has a unique subgroup of order 2n, so we conclude that hb,Z2i is normal in G. This implies that haihb,Z2i = ha, b,Z2i ◆ ha, bi = G, so |G|  |a| · |hb,Z2i| = 2 · 2n = 4n. This contradicts the fact that |G| = 4np. Subcase 7.3.1. Assume Z2 ✓ hbi. Combining this assumption with the above claim, we see that G0 ✓ hbi. This implies hbi / G, so G = hai n hbi. Since |a| = 2, this implies that Cay(G; a, b) is a generalized Petersen graph. Then the main result of [1] tells us that Cay(G; a, b) has a hamiltonian cycle. Subcase 7.3.2. Assume Z2 * hbi. Since hb,G0i is abelian, gcd(n, p) = 1, and Z2 * hbi, we may write hb,G0i = Z2 ⇥ Zp ⇥ Zn. Then G = hain (Z2 ⇥ Zp ⇥ Zn), and we may assume b = (0, 1, 1) and [a, b] = (1, 2, 0). For G = G/hb2i = G/(Zp ⇥ 2Zn), it is straightforward to check that (a, b)4#, b1 is a hamiltonian cycle in Cay(G; a, b) whose voltage is (0,2, 2). (This hamiltonian cycle is taken from the final paragraph of Case 1 of the proof of [3, Prop. 6.1].) This voltage generates Zp ⇥ 2Zn (since gcd(p, n) = 1), so the Factor Group Lemma (2.8) applies. 8 Cases with |a| = 2 and #S = 3 Assumption 8.1. In this section, we assume S = {a, b, c}, and |s| = 2, for all s 2 S, such that s does not centralize G0. We also assume Case 4.1 does not apply. (So |s| = 2.) In particular, we have |a| = 2. Note that a /2 hbi. (If a 2 hbi, then b, like a, does not centralize G0, so our assumption implies |b| = 2. Then a = b, contradicting the fact that Case 4.1 does not apply.) 20 Art Discrete Appl. Math. 1 (2018) #P1.04 Notation 8.2. Let n = |b| = |ha, bi : hai| 2 and ` = |G : ha, bi| = |G|/(2n) 2. The last inequality is because the irredundance of S implies c /2 ha, bi (see Corollary 3.5). Case 8.3. Assume |b| = 3. Proof. Since |b| 6= 2, Assumption 8.1 implies that b centralizes G0. Also, since |b| is odd, Corollary 3.8 implies that [a, b] and [b, c] project trivially to Z2, so [a, c] must project non- trivially (and ` must be even). We have the following hamiltonian path in Cay G/hai;S : L = (c`1, b, c(`1), b, c`1). Then C = (L, a, L1, a) is a hamiltonian cycle in Cay(G;S). Since `1 is odd, it is easy to see that Z2 ✓ h⇧Ci. Since C contains both [c`2](c, b, c1) and [c`1ab](b1), Lemma 2.12 (with s = c, t = b, u = a, and h = c`1a) provides a hamiltonian cycle C 0, such that (⇧C)1(⇧C 0) is conjugate to [t1, u] [s, t1]u = [b1, a] [c, b1]a = [a, b] [c, b]. This is an element of Zp. If it generates Zp, then either ⇧C or ⇧C 0 generates G0, so the Factor Group Lemma (2.8) applies. Thus, we may assume [a, b] [c, b] is trivial. Since Zp ✓ h[a, b]i (see (3.3B)), this implies that [c, b] is nontrivial. So we may assume that c does not centralize Zp (for otherwise replacing c with c1 would replace [c, b] with [c, b]1, which would not cancel [a, b]). Now, Assumption 8.1 implies |c| = 2, so we have the hamiltonian cycle C0 = (b 2 , a, b 2 , c, a, b, a, b, a, c), in Cay(G;S). This contains both the path [bac](a, b, a) and the edge [b](b), so applying Lemma 2.12 (with s = a, t = b, u = c, and h = b) provides a hamiltonian cycle C1, such that ⇧C0 1 ⇧C1 is conjugate to [u, t1] [s, t1]u = [c, b1] [a, b1]c. This is not equal to [a, b] [c, b] (which is trivial), because [a, b1]c = [a, b], but [c, b1] = [c, b]1 6= [c, b]. So ⇧C0 1 ⇧C1 is nontrivial, and therefore generates Zp. Since a straightfor- ward calculation shows that Z2 is contained in h⇧C0i, this implies that either ⇧C0 or ⇧C1 generates G0, so the Factor Group Lemma (2.8) applies. Case 8.4. Assume ` = 2. Proof. We may assume |b| 4, for otherwise either |b| = 2, so Theorem 2.3 applies (because |G| = 16p), or |b| = 3, so Case 8.3 applies. Let L = (a, b, a, bn2, a, b(n3)) and C = (L, c, L1, c1), so L is a hamiltonian path in Cay ha, bi; a, b and C is a hamiltonian cycle in Cay(G;S). Subcase 8.4.1. Assume [a, c] and [a, b][b, c] are not both in Zp. A straightforward calcula- tion (using Lemma 3.6) shows that ⇧C ⌘ [a, c] (mod Zp). If this is in Zp, then, by as- sumption, [a, b][b, c] /2 Zp, so applying Lemma 2.12 to the paths [e](a, b, a) and [abc](b1) in C (so s = a, t = b, u = c, and h = ac) yields a hamiltonian cycle C 0, such that ⇧C 0 projects nontrivially to Z2. Therefore, we have a hamiltonian cycle (either C or C 0) whose voltage is not in Zp. D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 21 Now, since |b| 4, we see that C (and also C 0) contains the path [b2ac](b, a, b1) and [ac](a). Furthermore, we know that [b, a][b, a]b is a nontrivial element of Zp (because b does not invert [a, b]). Therefore, Lemma 2.12 (with s = b, t = a, u = b, and h = ac) yields a hamiltonian cycle C1 (or C 01) whose voltage generates G0, so the Factor Group Lemma (2.8) applies. Subcase 8.4.2. Assume [a, c] and [a, b][b, c] are both in Zp. Since [a, c], [a, b], and [b, c] generate G0, they cannot all be in Zp, so this assumption implies that neither [a, b] nor [b, c] is in Zp. Also, we may assume h[a, c]i = Zp, for otherwise [a, c] = e, so we could apply Lemma 2.13 with s = c. We have the following hamiltonian cycle in Cay(G;S): C0 = (b n1 , c, b (n2) , a, b n2 , c 1 , b (n1) , c, a, c 1). Then ⇧C0 = b n1 c b (n2) ab n2 c 1 b (n1) cac 1 = bn1c a[a, b]n2 c 1 b (n1) cac 1 = ([a, b](n2))c · bn1(cac1)b(n1)(cac1) = ([a, b](n2))c · [b, cac1](n1) = ([a, b](n2))c · [b, a](n1) (cac1 2 aG0 and G0 ✓ CG(b)) = ([a, b](n2))c · [a, b]n1. If c centralizes Zp, then ⇧C0 = [a, b] generates G0, so the Factor Group Lemma (2.8) applies. We may now assume c does not centralize Zp. Then Assumption 8.1 tells us that c in- verts Zp, so ⇧C0 = [a, b]2n3 (and |c| = 2). Hence, we may assume 2n ⌘ 3 (mod p), for otherwise ⇧C0 generates G0, so the Factor Group Lemma (2.8) applies. We now consider the following hamiltonian cycle in Cay(G;S): C⇤ = (b n3 , c, b (n4) , a, b n4 , c 1 , b (n3) , c, (b1, c)2, a, (c, b)2, c1). We have ⇧C⇤ = b n3 c b (n4) ab n4 c 1 b (n3) c (b1c)2a(cb)2 c 1 . Since cb inverts G0, we know that (b1c)2a(cb)2 = a, so ⇧C⇤ is exactly the same as the voltage of C0, but with n replaced by n 2; that is, ⇧C⇤ = [a, b] 2(n2)3 = [a, b]2n7. Since 2n ⌘ 3 (mod p), we have 2n 7 ⌘ 3 7 = 4 6⌘ 0 (mod p), so ⇧C⇤ generates G0, so the Factor Group Lemma (2.8) applies. Case 8.5. Assume |b| 6= 3 and ` 6= 2. 22 Art Discrete Appl. Math. 1 (2018) #P1.04 Proof. Since ` 6= 2, we know |c| > 2, so c must centralize G0 (by Assumption 8.1). Also, Corollary 3.8 implies that |b| and ` cannot both be odd. • If |b| is odd (so ` is even), let L = c `1 , b, c 1 , b, c, b, (bn4, c1, b(n4), c1)`/2#, b1, c`3, b1, c(`3) . • If |b| is even, let L = c `1 , b n1 , c 1 , (c(`2), b1, c`2, b1)(n2)/2, c(`2) . In either case, L is a hamiltonian path in Cay G/hai; {b, c} from e to b. Now, let C = (L, a, L1, a) and (g, ✏) = ( (c`1,1) if |b| = 2 or |b| is odd, (ab2, 1) if |b| > 2 and |b| is even, so C is a hamiltonian cycle in Cay(G;S) that contains the paths [bc](c1, a, c), [ca](c1, a, c), [g](b), and [gbac✏](c✏, b1, c✏). Note that [bc](c1, a, c) contains [b](a) and that [ca](c1, a, c) contains [a](a). Also note that all of these paths are vertex-disjoint (except for the vertices ac and {abc} when |b| = 2 and ` = 3). We introduce some terminology: • Applying Lemma 2.12 to the oriented paths [ca](c1, a, c) and [b](a) (so s = c1, t = a, u = b, and h = ab) will be called the “a-transform.” This multiplies the voltage by a, where a = [a, b1][c, a]. • Applying Lemma 2.12 to the oriented paths [g](b) and [gbac✏](c✏, b1, c✏) (so s = c ✏, t = b1, u = a, and h = gb) will be called the “b-transform.” This multiplies the voltage by a conjugate of b, where b = [b, a][b, c✏]. Subcase 8.5.1. Assume precisely one of a and b is in Zp. Write {a, b} = {x, y}, such that x 2 Zp and y /2 Zp. We may assume hxi = Zp (by replacing c with its inverse, if necessary). Choose C 0 to be either C or the y-transform of C, such that ⇧C 0 projects nontrivially to Z2. Then choose C 00 to be either C 0 or the x-transform of C 0, such that ⇧C 00 generates G0, so the Factor Group Lemma (2.8) applies. Subcase 8.5.2. Assume a and b are both in Zp. Since [a, b], [a, c], and [b, c] cannot all be in Zp, this assumption implies that none of them are in Zp. Therefore, since the path L has odd length, we see that ⇧C has nontrivial projection to Z2. We may assume (by replacing c with its inverse, if necessary), that a has nontrivial projection to Zp, so hai = Zp. Therefore, by choosing C 0 to be either C or the a-transform of C, such that ⇧C 0 generates G0, we may apply the Factor Group Lemma (2.8). Subcase 8.5.3. Assume neither a nor b is in Zp, and b centralizes G0. Note that the sum of the exponents of the occurrences of b in L is 1, and the sum of the exponents of the occurrences of c is 0. Therefore, since b and c centralize G0, Lemma 3.6 implies that ⇧C = [a, b]. Hence, we may assume [a, b] 2 Zp (for otherwise h⇧Ci = G0, so the Factor Group Lemma (2.8) applies). Then, by the assumption of this subcase, we conclude that D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 23 [a, c] /2 Zp. So we may assume h[a, c]i = Z2, for otherwise b and c could be interchanged, resulting in a situation in which [a, b] /2 Zp, and which has therefore already been covered. Also, since [a, b] 2 Zp and [a, c] /2 Zp, Corollary 3.8 tells us that ` is even (and recall that ` 6= 2). Since [a, b] is a nontrivial element of Zp, and b centralizes G0, we see from Corollary 3.7 that |b| is divisible by p. Therefore, |b| 6= 2, so we may assume |b| > 2 (for otherwise Case 4.1 applies with s = b and t = b1). Since |b| 6= 3 (by the assumption of this case), this implies n = |b| 4, so we may let L0 = c `1 , b, c (`1) , b 2 , (bn4, c, b(n4), c)`/2#, b1, c(`2) , so L0 is a hamiltonian path from e to b2c in Cay G/hai; {b, c} . Note that the sum of the exponents of the occurrences of b in L is 2, and the sum of the exponents of the occurrences of c is 1. Therefore, since b and c centralize G0, Lemma 3.6 implies ⇧(L0, a, L10 , a) = [a, b]2[a, c]. This generates G0, so the Factor Group Lemma (2.8) applies. Subcase 8.5.4. Assume neither a nor b is in Zp, and b does not centralize Zp. From Assumption 8.1, we know b = 2 (so b must invert G0). We may assume [a, c] 2 Z2, for otherwise Case 8.4 could be applied by interchanging b and c. Then we may assume [a, c] is the nontrivial element of Z2, for otherwise the assumption that a /2 Zp implies h[a, b]i = G0, so ha, bi /G, and then Lemma 2.13 applies with s = c. By applying the same argument, with a and b interchanged, we may assume [b, c] is also the nontrivial element of Z2. This implies [a, b] 2 Zp, since b /2 Zp. Note that, since a and b both have order 2 (and invert G0), the image of ha, bi in G/Z2 is the dihedral group of order 2p. Also, the preceding two paragraphs imply that c is in the center of G/Z2. Therefore, we have the following hamiltonian cycle in Cay G/Z2;S : C = c, (c`2, a, c(`2), b)p#, c1, (a1, b1)p# . Since [a, b] projects trivially to Z2, Corollary 3.8 implies that ` is even, so, calculating modulo Zp, we have ⇧C = c(c`2ac(`2)b)pb1c1(a1b1)pb ⌘ c(ab)pb1c1(a1b1)pb ✓ ` 2 is even, so c`2 is central modulo Zp ◆ ⌘ z2p1(ab)pb1(a1b1)pb ✓ letting z = [a, c] = [b, c] be the nontrivial element of Z2 ◆ ⌘ z (z2 = e and [a, b] 2 Zp). Since this generates Z2, the Factor Group Lemma (2.8) applies. 9 Cases with |a| = 2 and #S 4 Assumption 9.1. In this section, we assume • #S 4, and • |s| = 2, for all s 2 S, such that s does not centralize G0. 24 Art Discrete Appl. Math. 1 (2018) #P1.04 We also assume Case 4.1 does not apply. (So |s| = 2.) Furthermore, we assume b /2 hai (otherwise, Case 4.1 applies). Then it is easy to see that we also have a /2 hbi. Outline. This final section of the proof is longer than the others, so here is an outline of the cases and subcases that it considers. 9.4: Assume no element of S centralizes G0. 9.4.1: Assume #S 5. 9.4.2: Assume #S = 4. 9.5: Assume there exists s 2 S, such that [a, s] /2 Zp, and, in addition, either s = b, or b centralizes G 0 , or Zp ✓ hS r {a}i0. 9.5.1: Assume Zp * hS r {a}i0. 9.5.2: Assume Zp ✓ hS r {a}i0. 9.6: Assume b centralizes G0. 9.6.1: Assume there exists c 2 S, such that [c, b] /2 Zp. 9.6.2: Assume [c, b] 2 Zp for all c 2 S. 9.7: Assume that none of the preceding cases apply. Since Case 9.4 does not apply, some element c of S centralizes G0. 9.7.1: Assume h[s, c]i 6= Z2, for some s 2 S r {c}. 9.7.2: Assume h[s, c]i = Z2, for all s 2 S r {c}. Notation 9.2. Let n = |b| and ` = |G : ha, bi| = |G|/(2n). Note 9.3. The irredundance of S implies S r {a, b} is an irredundant generating set for G/ha, bi (see Corollary 3.5), so ` 4. Case 9.4. Assume no element of S centralizes G 0 . Proof. From Assumption 9.1, we see that every element of S inverts G0 (and has order 2). We may assume no two elements of S commute, for otherwise it is not difficult to see that Lemma 2.13 applies. Let c, d 2 S r {a, b}, and let = [a, b] [a, c]. We claim that we may assume /2 Z2, by permuting b, c, d. To this end, first note that if 2 Z2, then Zp ✓ h[a, c]i, so there is no harm in putting c into the role of b. Now, let us suppose [a, b][a, c], [a, c][a, d], and [a, d][a, b] are all in Z2. Then [a, b] ⌘ [a, c]1 ⌘ [a, d] ⌘ [a, b]1 (mod Z2), which contradicts the fact that [a, b] /2 Z2 (and p is odd). Let C = (c, a, c, b)2#, d 2 , so C is a hamiltonian cycle in Cay ha, b, c, di; {a, b, c, d} that contains the vertex-disjoint paths [e](c, a, c), [abc](a), [bd](c, a, c), and [acd](a). Applying Lemma 2.12 to the paths D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 25 [e](c, a, c) and [abc](a) (so s = c, t = a, u = b, and h = bc) will multiply the voltage by . Applying Lemma 2.12 to the other two paths [bd](c, a, c) and [acd](a) (so s = c, t = a, u = b, and h = cd) will also multiply the voltage by (because bc and cd both centralize G0). Therefore, applying Lemma 2.12 twice yields a hamiltonian cycle C 00, such that (⇧C)1(⇧C 00) = 2, which is a generator of Zp. Subcase 9.4.1. Assume #S 5. If there exist s, t 2 S, such that s /2 {a, b, c}, and [s, t] /2 Zp, then the preceding paragraph implies that Lemma 3.15(2) applies. Thus, we may assume that the preceding condition does not apply (for any legitimate choice of a, b, and c). Fix two elements x, y 2 Sr{a, b, c}. The failure of the condition im- plies [x, S] ✓ Zp. In particular, [x, y] must be a generator of Zp (because no two elements of S commute), so we may let {x, y} play the role of {a, b}. So we may let {x, y, b, c} play the role of {a, b, c, d}. Then, since a /2 {x, y, b, c}, the failure of the condition implies [a, S] ✓ Zp. Similarly, [b, S] and [c, S] are also in Zp. So [s, t] ✓ Zp for all s, t 2 S. This contradicts the fact that h[S, S]i = G0 * Zp. Subcase 9.4.2. Assume #S = 4. For convenience, in this subcase (and only in this subcase), we drop our standing assumption that h[a, b]i contains Zp. Instead, choose b, d 2 S, such that [b, d] projects nontrivially to Z2. A straightforward calculation (using the fact that a, b, c, and d all invert G0) shows that ⇧C = [c, d]4[d, a]2[d, b]. Since [d, b] projects nontrivially to Z2, but [c, d]4 and [d, a]2 have even exponents, so they obviously do not, we see that Z2 ✓ h⇧Ci. Therefore, we may assume ⇧C 2 Z2, for otherwise the Factor Group Lemma (2.8) applies. We may assume 2 Z2, for otherwise applying Lemma 2.12 twice (as in the paragraph immediately before Subcase 9.4.1) yields a hamiltonian cycle whose voltage generates G0, so the Factor Group Lemma (2.8) applies. By the definition of , this means [a, b][a, c] 2 Z2. And we may assume the same is true when b and d are interchanged, which means [a, d][a, c] 2 Z2. So [a, b] ⌘ [a, c]1 ⌘ [a, d] (mod Z2). By interchanging a and c, we conclude that we may also assume [c, b] ⌘ [c, a]1 ⌘ [c, d] (mod Z2). So [c, d] ⌘ [c, a]1 = [a, c] ⌘ [a, d]1 = [d, a] (mod Z2). Therefore [d, a]6[d, b] = [d, a]4[d, a]2[d, b] ⌘ [c, d]4[d, a]2[d, b] = ⇧C ⌘ 0 (mod Z2). If p 6= 3, then, since we may assume the same is true when we interchange a and c, we conclude that [d, c] ⌘ [d, a] (mod Z2). Since we also have [c, d] ⌘ [d, a] (mod Z2), we conclude that [c, d] and [a, d] are in Z2. This implies [b, d] /2 Z2 (since d does not centralize Zp, and is therefore not in the center of G/Z2), so ⇧C = [c, d]4[d, a]2[d, b] ⌘ e4e2[d, b] = [d, b] 6⌘ 0 (mod Z2). This contradicts the fact that ⇧C 2 Z2. 26 Art Discrete Appl. Math. 1 (2018) #P1.04 We now assume p = 3. Then the equation [d, a]6[d, b] ⌘ 0 (mod Z2) implies [d, b] 2 Z2. This conclusion came from assuming only that [d, b] /2 Zp. Therefore, for all s, t 2 S, the commutator [s, t] must be in either Z2 or Zp. However, [a, b] ⌘ [c, a] ⌘ [a, d] ⌘ [b, c] ⌘ [d, c] (mod Z2), and [a, b] /2 Z2. Therefore, we conclude all five of these other commutators are in Zp. (Therefore, the stated congruences between these commutators are actually equalities.) Now, interchanging a $ b and c $ d in C yields a hamiltonian cycle C⇤, such that ⇧C⇤ = [d, c]4[c, b]2[c, a] = [d, c][b, c][c, a] = [c, a]3 = e (because p = 3). Let ⇤ = [b, a] [b, d], so ⇤ is obtained from = [a, b][a, c] by inter- changing a $ b and c $ d. Then, since applying Lemma 2.12 to C can multiply the voltage by = [a, b] [a, c], we know that applying Lemma 2.12 to C⇤ can multiply the voltage by ⇤, which generates G0. So the Factor Group Lemma (2.8) applies. Case 9.5. Assume there exists s 2 S, such that [a, s] /2 Zp, and: either s = b, or b centralizes G0, or Zp ✓ hS r {a}i0. Proof. Let S0 = S r {a}. Note that the irredundance of S implies a /2 hS0iZ2 (see Lemma 3.4). Subcase 9.5.1. Assume Zp * hS0i0. If [a, b] /2 Zp, we assume that s = b. Let g = ( s if [s, a] /2 Z2, sb 2 if [s, a] 2 Z2. Note that h[g, a]i = G0. Let H⇤ = hS0iZ2/Z2. From the assumption of this subcase, we know that H⇤ is abelian. Therefore, Corollary 2.11 provides a hamiltonian path L = (si)ri=1 in Cay(H⇤;S0), such that s1s2 · · · sr 2 gZ2. Then (L1, a, L, a) is a hamiltonian cycle in Cay(G;S), and ⇧C = [s1s2 · · · sr, a] 2 [gZ2, a] = {[g, a]} (since Z2 is in the center of G). This voltage generates G0, so the Factor Group Lemma (2.8) applies. Subcase 9.5.2. Assume Zp ✓ hS0i0. Suppose w, x, y 2 S±1 r {a}, such that hwi ( hw, xi ( hw, x, yi. (9.5A) It is easy to construct a hamiltonian cycle C0 in Cay(hS0i;S0), such that C0 contains the oriented paths [hw1y1](w, x,w1) and [hx](x1), for some h 2 G. Furthermore, if either x /2 {s±1} or |G| > 16, (9.5B) then, for some ✏ 2 {±1}, it is not difficult to arrange that the hamiltonian cycle C0 contains the oriented edge [s✏](s✏), and that this edge is not in either of the above-mentioned paths. D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 27 Applying Lemma 2.12 to the first two paths (so s = w, t = x, and u = y) yields a hamiltonian cycle C1, such that (⇧C0)1(⇧C1) is conjugate to [x1, y] [w, x1]y . Re- moving the edge [s✏](s✏) yields hamiltonian paths C0# and C1# from e to s✏. From Lemma 3.4 and the assumption of this subcase, we see that hS0i 6= G. So C + 0 = C0#, a, (C0#) 1 , a and C+1 = C1#, a, (C1#) 1 , a are hamiltonian cycles in Cay(G;S). For k = 0, 1, we have ⇧C+k = ⇥ (⇧Ck)s ✏ 1 , a ⇤ . Since ⇧Ck 2 G0, and G0 is central modulo Zp (and from the choice of s), we have ⇧C+k ⌘ [s ✏ , a] 6⌘ e (mod Zp). Furthermore, if [x1, y] [w, x1]y projects nontrivially to Zp, then (⇧C+0 )1(⇧C+1 ) does not centralize a modulo Z2, so ⇧C+0 and ⇧C+1 are not both in Z2. This implies that ⇧C+k generates G 0 for some k, so the Factor Group Lemma (2.8) applies. Therefore (after replacing x1 with x for simplicity), we may assume [w, x]y [x, y] 2 Z2 for all w, x, y 2 S±1 r {a} that satisfy (9.5A) and (9.5B). (9.5C) We will show that this leads to a contradiction. Assume, for the moment, that b centralizes G0. Then n = |b| > 2 (because Corol- lary 3.7 implies that |b| 6= 2), so |G| = 2n` > 2 · 2 · 4 = 16. Therefore (9.5B) is automatically satisfied. Let x, y 2 S0 r {b}, such that x 6= y. We see from Note 9.3 that (9.5A) is satisfied for w = b±1, so (9.5C) tells us [b, x]y [x, y] and [b1, x]y [x, y] are both in Z2. However, we also know that [b1, x] = [b, x]1 (because we are assuming in this paragraph that b centralizes G0). Therefore [b, x]y ⌘ [x, y]1 ⌘ [b1, x]y = [b, x]1 y (mod Z2), so [b, x] 2 Z2 (for all x 2 S0). Then, since [b, x]y [x, y] 2 Z2, we conclude that [x, y] 2 Z2, for all x, y 2 S0. This contradicts the assumption of this subcase. Now assume b does not centralize G0. We may assume Case 9.4 does not apply, so G0 is centralized by some t 2 S (and t 6= b). Let w, x 2 S0 r {t} with w 6= x. Combining the irredundance of S with the fact that t 6= b implies that (9.5A) is satisfied for y = t±1 (unless w = x, when Case 4.1 applies). We may assume x 6= s (by interchanging w and x, if necessary), so (9.5B) is satisfied. Then (9.5C) tells us [w, x]t [x, t] and [w, x]t 1 [x, t1] are both in Z2. Since t centralizes G0, this implies [x, t] ⌘ [x, t1] = [x, t]1 (mod Z2), so [x, t] 2 Z2 (for all x 2 S0). Since [w, x]t [x, t] 2 Z2, this implies [w, x] 2 Z2 (for all w, x 2 S0). This contradicts the assumption of this subcase. Case 9.6. Assume b centralizes G 0 . 28 Art Discrete Appl. Math. 1 (2018) #P1.04 Proof. We consider two subcases. Subcase 9.6.1. Assume there exists c 2 S, such that [c, b] /2 Zp. We use some of the arguments of Case 8.5. We may assume [a, s] 2 Zp for all s 2 S. (Otherwise, Case 9.5 applies, because b centralizes G0.) Therefore c 6= a. Let L = (si)ri=1 be a hamiltonian path from e to b in Cay G/hai;S r {a} , such that s1 = c = s1r , and L contains a path of the form [gc✏](c✏, b, c✏) (for some , ✏ 2 {±1}) that is vertex-disjoint from {e, c, b, bc}. Now let C = (L, a, L1, a). Then C contains vertex-disjoint paths of the form [b](a), [ca](c1, a, c), [gc✏](c✏, b, c✏), and [gab](b). • Applying Lemma 2.12 to [b](a) and [ca](c1, a, c) (so s = c1, t = a, u = b, and h = ab) will be called the “a-transform.” It multiplies the voltage by a = [b, a][a, c 1]. • Applying Lemma 2.12 to [gc✏](c✏, b, c✏) and [gab](b) (so s = c✏, t = b , u = a, and h = ga) will be called the “b-transform.” It multiplies the voltage by a conjugate of b = [a, b][c ✏ , b]. Since [a, b], [a, c] 2 Zp and [b, c] /2 Zp we know a 2 Zp and b /2 Zp. Also, we may also assume a is nontrivial (by replacing b with b1 if necessary). Therefore, the argument of Subcase 8.5.1 applies. Namely, choose C 0 to be either C or the b-transform of C, such that ⇧C 0 projects nontrivially to Z2. Then choose C 00 to be either C 0 or the a-transform of C 0, such that ⇧C 00 generates G0, so the Factor Group Lemma (2.8) applies. Subcase 9.6.2. Assume [c, b] 2 Zp for all c 2 S. Choose c, d 2 S, such that [c, d] /2 Zp. Assuming that Case 9.5 and Subcase 9.6.1 do not apply, we have [s, t] 2 Zp for all s 2 {a, b} and t 2 S. Therefore, c, d /2 {a, b}, and the element = [a, b][d1, a] is in Zp, and we may assume (by replacing b with its inverse, if necessary) that generates Zp. Let S0 = {a, b, d}, and choose a hamiltonian cycle C0 in Cay hS0i;S0 that con- tains the oriented paths [d](d1, a, d) and [ab](a), and has at least two edges labelled x±1, for every x 2 S0. Lemma 2.12 (with s = d1, t = a, u = b, and h = b) provides a hamiltonian cycle C1, such that (⇧C0)1(⇧C1) is conjugate to , and therefore gener- ates Zp. Furthermore, C1 contains all of the oriented edges of C0 that are not in these two above-mentioned paths, so Lemma 3.15(2) applies (with g = b and t = d). Case 9.7. Assume that none of the preceding cases apply. Proof. This implies that: #1. [a, b] 2 Zp. (Otherwise, Case 9.5 applies.) #2. If s 2 S, and there exists t 2 S, such that t inverts G0 and Zp ✓ h[s, t]i, then s inverts G 0 . (If s does not invert G0, then we see from Assumption 9.1 that s cen- tralizes G0, so Case 9.6 applies with s and t in the roles of b and a, respectively.) D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 29 #3. There exists c 2 S, such that c centralizes G0. (Otherwise, Case 9.4 applies.) From (#2), we know [a, c] 2 Z2. Subcase 9.7.1. Assume h[s, c]i 6= Z2, for some s 2 S r {c}. Suppose, for the moment, that s centralizes G0. Then Lemma 3.6 implies ⇥ a, [s, c] ⇤ = ⇥ [a, s], [a, c] ⇤ = e (because G0 is abelian), so [s, c] projects trivially to Zp. Since h[s, c]i 6= Z2, we conclude from this that [s, c] = e, so Lemma 2.16 applies. We may now assume s does not centralize G0, so there is no harm in assuming that s = a. Since (#2) implies that [a, c] 2 Z2, we see that [a, c] must be trivial. Let H = hSr{c}i. We may assume Z2 * H , for otherwise H / G, so Lemma 2.13 applies with s = c and t = a. Therefore, [x, y] 2 Zp for all x, y 2 S r {c}, but there is some d 2 S r {c}, such that [c, d] projects nontrivially to Z2. Similarly, we may assume Zp * hS r {a}i, for otherwise we have hS r {a}i / G, so Lemma 2.13 applies with s = a and t = c. This means [x, y] 2 Z2 for all x, y 2 S r {a}. In particular, since b and d are in both Sr{a} and Sr{c}, we must have [b, d] 2 Z2\Zp = {e}. Choose a hamiltonian cycle C0 in Cay H;S r {c} that contains the oriented paths [d](d1, b, d) and [ab](b). If we apply Lemma 2.12 to these paths (so s = d1, t = b, u = a, and h = a), then the voltage is multiplied by a conjugate of [b, a] [b, d1], which is a generator of Zp (since [a, b] generates Zp and [b, d] is trivial). Therefore, Lemma 3.15(1) applies with s = t = d and u = a. Subcase 9.7.2. Assume h[s, c]i = Z2, for all s 2 S r {c}. For convenience, let bG = G/Z2 and bH = hbSr{bc}i. Then | bH 0| = p is prime, so Theorem 1.1 provides a hamiltonian path L in Cay bH;S r {c} . Since bc is central in bG, there is a spanning subgraph of Cay( bG;S) that is isomorphic to the Cartesian product L ⇤ (bc`1), where ` = |G : hS r {c}i|. Since | bG| is even, it is easy to find a hamiltonian cycle C in L ⇤ (bc`1) (see Lemma 2.10), and this yields a hamiltonian cycle bC in Cay( bG;S). To complete the proof, we carry out a straightforward (and well-known) calculation to verify that ⇧ bC is nontrivial, so the Factor Group Lemma (2.8) applies. If we view the Cartesian product L⇤ (bc`1) as a grid of squares, then the interior of the hamiltonian cycle C is a union of squares of the grid. Graph theoretically, this means C is the connected sum of some number N of digons of the form [g](t, t1) (where t 2 S±1). Note that if C is an r-cycle (with r 2), then C #st (t, t1) is an (r + 2)-cycle. Therefore, since the length of C is | bG|, we have 2N = | bG| ⌘ 0 (mod 4), so N is even. Now, each 4-cycle in L⇤ (bc`1) is of the form [bg](s1, t1, s, t), where one of s and t is in {c±1}, and the other is in S±1 r {c±1}. This means that in any connected sum C #st [g](t, t1), one of s and t is in {c±1}, and the other is in S±1 r {c±1}. By the assumption of this subcase, we conclude that [s, t] = z, where z is the generator of Z2. Therefore ⇧C = ⇧ ⇣ [ bg1](t1, t11 ) # s2 t2 [ bg2](t2, t 1 2 ) # s3 t3 · · · # sN tN [cgN ](tN , t 1 N ) ⌘ ⌘ YN i=2 [si, ti] (Corollary 3.14 and ⇧(t, t1) = e) = zN1 6⌘ e (mod Zp) (N 1 is odd). 30 Art Discrete Appl. Math. 1 (2018) #P1.04 References [1] K. Bannai, Hamiltonian cycles in generalized Petersen graphs, J. Comb. Theory Ser. B 24 (1978), 181–188, doi:10.1016/0095-8956(78)90019-9. [2] C. C. Chen and N. F. Quimpo, On strongly Hamiltonian abelian group graphs, in: K. L. McA- vaney (ed.), Combinatorial Mathematics VIII, Springer, Berlin, volume 884 of Lecture Notes in Mathematics, pp. 23–34, 1981, doi:10.1007/bfb0091805, proceedings of the Eighth Australian Conference on Combinatorial Mathematics Held at Deakin University, Geelong, August 25 – 29, 1980. [3] S. J. Curran, D. Witte Morris and J. Morris, Cayley graphs of order 16p are hamiltonian, Ars Math. Contemp. 5 (2012), 185–211, http://amc-journal.eu/index.php/amc/ article/view/207. [4] E. Durnberger, Connected Cayley graphs of semi-direct products of cyclic groups of prime order by abelian groups are Hamiltonian, Discrete Math. 46 (1983), 55–68, doi:10.1016/ 0012-365x(83)90270-4. [5] E. Durnberger, Every connected Cayley graph of a group with prime order commutator group has a Hamilton cycle, in: B. R. Alspach and C. D. Godsil (eds.), Cycles in Graphs, North- Holland, Amsterdam, volume 115 of North-Holland Mathematics Studies, pp. 75–80, 1985, doi:10.1016/s0304-0208(08)72997-9, papers from the workshop held at Simon Fraser Univer- sity, Burnaby, B.C., July 5 – August 20, 1982. [6] E. Ghaderpour and D. Witte Morris, Cayley graphs of order 27p are hamiltonian, Int. J. Comb. 2011 (2011), Article ID 206930 (16 pages), doi:10.1155/2011/206930. [7] E. Ghaderpour and D. Witte Morris, Cayley graphs of order 30p are hamiltonian, Discrete Math. 312 (2012), 3614–3625, doi:10.1016/j.disc.2012.08.017. [8] E. Ghaderpour and D. Witte Morris, Cayley graphs on nilpotent groups with cyclic commutator subgroup are hamiltonian, Ars Math. Contemp. 7 (2014), 55–72, http://amc-journal. eu/index.php/amc/article/view/280. [9] C. Godsil and G. Royle, Algebraic Graph Theory, volume 207 of Graduate Texts in Mathemat- ics, Springer-Verlag, New York, 2001, doi:10.1007/978-1-4613-0163-9. [10] J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1987. [11] K. Keating and D. Witte, On Hamilton cycles in Cayley graphs in groups with cyclic com- mutator subgroup, in: B. R. Alspach and C. D. Godsil (eds.), Cycles in Graphs, North- Holland, Amsterdam, volume 115 of North-Holland Mathematics Studies, pp. 89–102, 1985, doi:10.1016/S0304-0208(08)72999-2, papers from the workshop held at Simon Fraser Univer- sity, Burnaby, B.C., July 5 – August 20, 1982. [12] K. Kutnar, D. Marušič, D. Witte Morris, J. Morris and P. Šparl, Hamiltonian cycles in Cayley graphs whose order has few prime factors, Ars Math. Contemp. 5 (2012), 27–71, http:// amc-journal.eu/index.php/amc/article/view/177. [13] D. Marušič, Hamiltonian circuits in Cayley graphs, Discrete Math. 46 (1983), 49–54, doi: 10.1016/0012-365x(83)90269-8. [14] E. Schenkman, Group Theory, Robert E. Krieger Publishing Co., Huntington, New York, 1975, corrected reprint of the 1965 edition. [15] D. Witte and J. A. Gallian, A survey: Hamiltonian cycles in Cayley graphs, Discrete Math. 51 (1984), 293–304, doi:10.1016/0012-365x(84)90010-4. D. W. Morris: Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian 31 [16] D. Witte Morris, Odd-order Cayley graphs with commutator subgroup of order pq are hamil- tonian, Ars Math. Contemp. 8 (2015), 1–28, http://amc-journal.eu/index.php/ amc/article/view/330. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.05 https://doi.org/10.26493/2590-9770.1242.809 (Also available at http://adam-journal.eu) Most rigid representations and Cayley index ⇤ Joy Morris , Josh Tymburski Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada Received 27 March 2017, accepted 3 January 2018, published online 6 February 2018 Abstract For any finite group G, a natural question to ask is the order of the smallest possible automorphism group for a Cayley graph on G. A particular Cayley graph whose auto- morphism group has this order is referred to as an MRR (Most Rigid Representation), and its Cayley index is a numerical indicator of this value. Study of GRRs showed that with the exception of two infinite families and thirteen individual groups, every group admits a Cayley graph whose MRR is a GRR, so that the Cayley index is 1. The full answer to the question of finding the smallest possible Cayley index for a Cayley graph on a fixed group was almost completed in previous work, but the precise answers for some finite groups and one infinite family of groups were left open. We fill in the remaining gaps to completely answer this question. Keywords: Cayley graph, Cayley index, GRR, MRR, automorphisms. Math. Subj. Class.: 05C25 1 Introduction All groups and graphs in this paper are finite. All of our graphs are simple, undirected, and have no loops. A Cayley graph = Cay(G,S) where S ✓ G with S = S1 and 1 /2 S, is the graph whose vertices are the elements of G, with (g, gs) 2 E() if and only if g 2 G and s 2 S. We refer to S as the connection set for . Let A = Aut(). Observe that LG, the left-regular representation of G, lies in A, so |G| divides |A|. ⇤The authors would like to thank the anonymous referee whose careful reading and generous advice helped us to correct some errors and historical inaccuracies, and improved the exposition of this paper. This research was supported in part by the Natural Science and Engineering Research Council of Canada, Grant RGPIN-2011-238552. E-mail addresses: joy.morris@uleth.ca (Joy Morris), josh.tymburski@uleth.ca (Josh Tymburski) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.05 Definition 1.1. The Cayley index c() of the Cayley graph = Cay(G,S), is |A : LG|. The Cayley index c(G) of the group G is minS✓G,S=S1 c(Cay(G,S)); that is, the lowest Cayley index of any Cayley graph on the group G. Definition 1.2. A Cayley graph = Cay(G,S) is a GRR (Graphical Regular Representa- tion) for G if c() = 1. Thus, groups that admit GRRs are precisely the groups whose Cayley index is 1. In order to completely characterise these groups, we require another definition. Definition 1.3. Let A be an abelian group of even order, and y an involution in A. Then the generalised dicyclic group Dic(A, y) is hA, xi where x /2 A, x2 = y, and x1ax = a1 for every a 2 A. Notice that under this definition, the generalised dicyclic group Dic(A, y) will be abelian if and only if A is an elementary abelian 2-group. The study of GRRs involved many researchers and papers. Some of the most influential work along the way appeared in [6, 7, 9]. Watkins [16] observed that there are two infinite families of graphs that cannot admit GRRs: generalised dicyclic groups, and abelian groups that are not elementary abelian 2-groups. Imrich [7] resolved the problem for abelian groups by classifying the elementary abelian 2-groups, finding exactly three that admit no GRR. Watkins in a series of papers, some with coauthor Nowitz [11, 12, 16, 17, 18], discovered ten nonabelian groups that admit no GRRs. Hetzel [5] proved that aside from the two infinite families noted by Watkins, and the thirteen small solvable groups (of order at most 32) found by Imrich, Nowitz, and Watkins, every solvable group admits a GRR. Godsil [3] showed that every non-solvable group admits a GRR. In the case where a group fails to admit a GRR, a natural question to ask is: what is the Cayley index of the group, and what is a Cayley graph on the group that has that Cayley index? The following terminology was coined in [10]. Definition 1.4. Let G be a group with c(G) > 1, and let = Cay(G,S) be a Cayley graph on G with c() = c(G). Then we say that is an MRR (Most Rigid Representation) for G. The bulk of this paper is divided into 4 sections. In Section 2, we describe the groups that do not admit a GRR but do not lie in either of the infinite families of groups that do not admit a GRR. For each of these groups, we find its Cayley index and an MRR. In Section 3, we find the Cayley index of every abelian group, and find MRRs for those groups whose Cayley index is greater than 2. In Section 4, we consider a subfamily of generalised dicyclic groups (specifically, the hamiltonian 2-groups), and show that the smallest two of these have Cayley index 16, while the rest have Cayley index 8. Finally, in Section 5, we find the Cayley index for every generalised dicyclic group that was not included in Section 4. Much of the work that we summarise in this paper was done in [10], but the authors of [10] left some gaps. Our paper fills all of these gaps, thus completing their work. Specifi- cally, we fill the following gaps. We examine the Cayley indices of the groups that do not lie in either of the infinite families; we give the Cayley indices for the four abelian groups for which they did not specify it (although they stated that these had been found by computer); we find the precise Cayley index for generalised dicyclic groups of order at most 96 (they bounded almost all of these by 4, but most in fact have Cayley index 2); and we find the J. Morris and J. Tymburski: Most rigid representations and Cayley index 3 Cayley indices for all hamiltonian 2-groups (they bounded these by 16, but almost all have Cayley index 8). Table 1 summarises this work, providing the Cayley index for every finite group. For a number of the small individual groups, we found MRRs using Sage [15] and its GAP package [14]. The Cayley index of any of the graphs we present can be easily checked via computer, using this or other appropriate software. Throughout this paper, Q8 = {±1,±i,±j,±k : ij = k = ji, i2 = j2 = k2 = 1} is the usual representation of the quaternion group of order 8. We use D2n for n 3 to rep- resent the dihedral group of order 2n. Four of the exceptional groups listed in Theorem 2.1 we denote by Hi for i 2 {1, 2, 3, 4}; a precise representation of each of these groups is given in Theorem 2.1. To represent some of our MRRs, we use cartesian products. For two graphs 1 and 2, the cartesian product of 1 with 2 is denoted by 1 ⇤2. It is the graph whose vertices are the elements of V (1)⇥ V (2), with (u1, v1) adjacent to (u2, v2) if and only if either u1 = u2 and v1 is adjacent to v2 in 2, or v1 = v2 and u1 is adjacent to u2 in 1. We say that a graph on more than one vertex is prime with respect to the cartesian product if ⇠= 1 ⇤2 implies that for some i 2 {1, 2}, i ⇠= and 2i has just one vertex. It is well-known that every graph has a unique prime factorisation as the cartesian product of prime graphs. We say that two graphs are relatively prime with respect to the cartesian product if they have no common factors in their prime factorisations. We sometimes simply refer to the graphs as prime or relatively prime. 2 Exceptional groups We begin by listing the 13 groups that do not admit a GRR but do not lie in either of the infinite families that do not admit GRRs. The following theorem is the end result of considerable work by a number of research- ers. Imrich [7] completed the abelian case (correcting an earlier error by Sabidussi [13] and Chao [2], who showed that no graph has a transitive abelian automorphism group, but overlooked the case of elementary abelian 2-groups). The construction given in [7] also has an error in the case of the elementary abelian 2-group of order 32; it is mentioned in [10] that this was pointed out and corrected by Alspach, Hell, Hetzel, and Lim, and a GRR for that group (due to Hetzel) appears in [10]. Watkins, alone and in joint work with Nowitz [11, 12, 16, 17, 18] found the other ten exceptional groups and proved in [12] that any nonabelian group whose order is coprime to 6 admits a GRR. Imrich [8] then showed that every nonabelian group whose order is odd and at least 37 · 54 admits a GRR. Hetzel [5, Satz 14.38] showed that the exceptions we have mentioned are the only solvable groups that fail to have GRRs, and Godsil [3] completed the result by showing that every nonsolvable group has a GRR. We therefore cite Godsil’s work for the final result, but attribute it to all of the researchers who made major contributions. Theorem 2.1 (Godsil, Hetzel, Imrich, Nowitz, and Watkins; see [3]). The following are the only groups that are neither generalised dicyclic nor abelian of exponent greater than 2, yet admit no GRR: • Z22, Z32, Z42; 4 Art Discrete Appl. Math. 1 (2018) #P1.05 • D6, D8, D10 where these represent the dihedral groups of orders 6, 8, and 10 (re- spectively); • A4, the alternating group of degree 4; • H1 := ha, b, c : a2 = b2 = c2 = 1, abc = bca = cabi; • H2 := ha, b : a8 = b2 = 1, bab = a5i; • H3 := ha, b, c : a3 = b3 = c2 = 1, ab = ba, (ac)2 = (bc)2 = 1i; • H4 := ha, b, c : a3 = c3 = 1, ac = ca, bc = cb, b1ab = aci; • Q8 ⇥ Z3, Q8 ⇥ Z4, where Q8 is the quaternion group of order 8. The groups listed in the first bullet are abelian, and their Cayley indices are given in Section 3. All of the remaining groups have Cayley index 2. Their Cayley index must be at least 2 by Theorem 2.1, since they admit no GRR. This was shown explicitly in [16, Theorem 2] Table 1: Cayley indices for all finite groups. Group Cayley index See Abelian groups Z2,Zn2 , n 5 1 [7], 1.2 of [10] Z32,Z4 ⇥ Z2 6 Lemma 2.7 of [10] Z42 8 Table 3 Z24 4 Table 3 Z4 ⇥ Z22 8 Table 3 Z23 8 Lemma 2.4 of [10] Z33 12 Table 3 all other abelian groups 2 Theorem 1 of [10] Hamiltonian 2-groups Q8 16 Lemma 2.6 of [10] Q8 ⇥ Z2 16 Section 4 Q8 ⇥ Zn2 , n 2 8 Proposition 4.8 Other generalised dicyclic groups Dic(Z6, 3) 4 Table 4 Dic(Z8, 4) 4 Table 4 Dic(Z10, 5) 4 Table 4 Dic(Z4 ⇥ Z2, (0, 1)) 4 Table 4 all other generalised dicyclic groups 2 Section 5, and Theorem 2 of [10] Exceptional groups D6, D8, D10 2 Section 2 of [10], or Table 2 A4 2 Table 2 Q8 ⇥ Z3, Q8 ⇥ Z4 2 Table 2 H1 of order 16 2 Table 2 H2 of order 16 2 Table 2 H3 of order 18 2 Table 2 H4 of order 27 2 Table 2 Every group not listed above 1 [3] J. Morris and J. Tymburski: Most rigid representations and Cayley index 5 for the dihedral groups in the second bullet. It was shown in [18, Proposition 3.7] for A4. For the groups H1 and H3, it was shown in [18, Proposition 5.3 and Theorem 2]. The group H2 was dealt with in [11, Theorem 2 or Proposition 3.1], and H4 in [12, Theorem 3]. Finally, Q8 ⇥ Z3 and Q8 ⇥ Z4 were addressed in [17, Theorem]. To show that the Cayley index of each is precisely 2, we present Table 2. For each group, we give the connection set for a Cayley graph on that group that has Cayley index 2. The Cayley indices of these graphs can be verified by hand or by computer. Table 2: MRRs for exceptional groups. Group G S such that c(Cay(G,S)) = 2 D2n = ha, b : a2 = bn = 1, aba = b1i, {a, ab} n 2 {3, 4, 5} A4 {(1 2 3)±1, (1 2)(3 4)} H1 = ha, b, c : a2 = b2 = c2 = 1, {a, b, c, (ab)±1} abc = bca = cabi H2 = ha, b : a8 = b2 = 1, bab = a5i {a±1, a±2, b} H3 = ha, b, c : a3 = b3 = c2 = 1, ab = ba, {a±1, c, ac, bc} (ac)2 = (bc)2 = ei H4 = ha, b, c : a3 = c3 = 1, ac = ca, bc = cb, {a±1, b±1, (a1b)±1, (bab1)±1} b 1 ab = aci Q8 ⇥ Z3 = hi, j, z : z3 = 1, iz = zi, jz = zji {±i, (iz)±1, (jz)±1} Q8 ⇥ Z4 = hi, j, z : z4 = 1, iz = zi, jz = zji {z±1,±i,±j, (iz)±1, (kz)±1} The MRRs listed in the first line of this table were also mentioned in [10]. 3 Abelian groups The Cayley index of every abelian group was determined in [10]. However, for a small number of these they stated only that the Cayley index had been found by Hetzel on com- puter, and cite a private communication. The known results on abelian groups are as fol- lows. Theorem 3.1 ([10, Theorem 1, Lemma 2.4, Lemma 2.7]). The only finite abelian groups with a Cayley index greater than 2 are: • Z32 and Z4 ⇥ Z2, for which the Cayley index is 6, with MRR K2 ⇤K2 ⇤K2 (the cube); • Z23, for which the Cayley index is 8, with MRR K3 ⇤K3; • Z42, Z4 ⇥ Z22, and Z24; and • Z33. In the rest of this section, we list the Cayley index for each of the last four groups together with an MRR for each group. The Cayley indices for these graphs and the fact that these are the Cayley indices for these groups can be verified by computer. 6 Art Discrete Appl. Math. 1 (2018) #P1.05 If A is an abelian group that we are presenting as being isomorphic to Zi1 ⇥ · · ·⇥Zik , then we let {z1, . . . , zk} be the canonical generating set for this group, so |zj | = ij . We present the Cayley index and an MRR for each group in Table 3. Table 3: MRRs for abelian groups not given in [10]. Group Cayley index Connection set for an MRR Z42 8 {z1, z2, z3, z4, z1z2, z1z3, z2z4} Z4 ⇥ Z22 8 {z±11 , z2, z3, (z1z2)±1, (z1z3)±1} Z24 4 {z±11 , z±12 , z21 , (z1z2)±1} Z33 12 {z±11 , z±12 , z±13 , (z1z2)±1, (z1z3)±1, (z2z3)±1} It may seem odd that c(Z42) > c(Z32). However, Lemma 4.4 does not apply here, because neither MRR for Z32 (K2 ⇤K2 ⇤K2 and its complement, K4 ⇤K2), is relatively prime to K2, which is the unique connected MRR for Z2. 4 The groups Q8 ⇥ Zn2 In this section we deal with a particular family of generalised dicyclic groups: groups of the form Q8 ⇥ Zn2 for some nonnegative integer n. Definition 4.1. A hamiltonian group is a nonabelian group all of whose subgroups are normal. A hamiltonian 2-group is a hamiltonian group whose order is a power of 2. It is well-known (see, for example, [4, Theorem 12.5.4]) that the hamiltonian 2-groups are precisely the groups of the form Q8 ⇥ Zn2 for some nonnegative integer n that we are considering in this section. We begin with three important results from [10]. Lemma 4.2 ([10, Lemma 2.6]). The group Q8 has Cayley index 16, with C4 ⇤K2 as an MRR. Lemma 4.3 ([10, Proposition 2.9]). Every group other than Z22, Z32, Z4, Z4 ⇥ Z2, and Z23 admits a connected MRR that is prime with respect to the cartesian product. Lemma 4.4 ([10, Lemma 2.8]). Let G1 and G2 be groups having connected MRRs that are relatively prime with respect to the cartesian product. Then c(G1 ⇥G2)  c(G1)c(G2). In fact, if 1 and 2 are connected MRRs for G1 and G2 (respectively) that are rel- atively prime with respect to the cartesian product, then c(1 ⇤2) = c(G1)c(G2) and 1 ⇤2 is a Cayley graph on G1 ⇥G2. The following observation is made in [10] and is implicit in their Theorem 2(b), which states that c(Q8 ⇥ Zn2 )  16 for every integer n 0. It can be deduced from Lemmas 4.2, 4.3, and 4.4, using the fact that c(Z2) = 1. Corollary 4.5. For every group G /2 {Z22,Z32,Z4,Z4 ⇥ Z2,Z23}, c(G⇥ Z2)  c(G). The following result is key to providing a lower bound for the Cayley index of every group Q8 ⇥ Zn2 . J. Morris and J. Tymburski: Most rigid representations and Cayley index 7 Proposition 4.6 ([1, Classification Theorem]). There are 8 permutations ' of the elements of G = Q8 ⇥ Zn2 that fix the identity, and have the property that for every g, h 2 G, '(gh) is either '(g)h, or '(g)h1. Corollary 4.7. The Cayley index of Q8 ⇥ Zn2 is at least 8 for every integer n 0. Proof. Fix n, and let G = Q8 ⇥ Zn2 . Let S be any inverse-closed subset of G \ {1G}, and let = Cay(G,S). Let ' be any of the 8 permutations given in Proposition 4.6. To prove this result, it will be sufficient to show that ' is an automorphism of . We know that for any g 2 G, g is adjacent to gs if and only if s 2 S. We also know that '(gs) is either '(g)s, or '(g)s1. Since S is inverse-closed, each of these is adjacent to '(g) if and only if s 2 S. Thus, ' is indeed an automorphism of . To complete this section, we note that C4 ⇤K2 ⇤K2 is an MRR for Q8 ⇥ Z2 with Cayley index 16, verified by computer. However, for Q8 ⇥ Z22, the Cayley index is 8, with MRR Cay(Q8⇥Z22, {±i,±j,±k,±iz1,±kz1z2, z1, z2}), where z1 and z2 are two distinct central involutions that do not lie in Q8. Thus, using Corollary 4.5 and Corollary 4.7 we are able to conclude the following. Proposition 4.8. For every integer n 2, the Cayley index of Q8 ⇥ Zn2 is 8. 5 Other generalised dicyclic groups Imrich and Watkins [10] showed that generalised dicyclic groups of order greater than 96 that are not of the form Q8 ⇥ Zn2 have Cayley index 2. Many of the ideas from their proof in fact apply to generalised dicyclic groups of smaller orders. We reproduce these key ideas here, without their assumptions on order. We generally need to find two elements that satisfy a number of conditions. We note that the condition a1 6= ya2 was not listed in [10] but is required; for this reason we provide a full proof of Lemma 5.4. Definition 5.1. Let Dic(A, y) be a generalised dicyclic group. We say that the 2-set {a1, a2} for a1, a2 2 A is a suitable pair of elements of Dic(A, y) if for every {i, j} = {1, 2} we have (i) a1 6= a2, ya2; (ii) a2i 6= 1, y; (iii) ai 6= a2j , ya2j ; and (iv) a1a2 6= 1, y. Lemma 5.2. Let D = Dic(hzi, zn) (the dicyclic group of order 4n), where |z| = 2n > 10. Then {z, z2} is a suitable pair for D. If = Cay(D, {z±1, x±1, (xz)±1, (xz2)±1}), where x 2 = zn, then hzi is invariant under Aut()1. Proof. We have y = zn. We verify the conditions for {z, z2} to be a suitable pair. Since n > 5, (i) and (ii) are satisfied; (iii) and (iv) are equally easy to check. Let ' 2 Aut()1 be arbitrary. It is straightforward to verify that when n > 4, zn is the unique vertex that has 6 common neighbours with 1, so '(zn) = zn. In fact, this shows that for any vertex v, vzn is uniquely determined as the vertex that has 6 common neighbours with v. Since the neighbours of 1 can be partitioned into three pairs of this 8 Art Discrete Appl. Math. 1 (2018) #P1.05 sort ({x, x1 = xzn}, {xz, xzn+1}, and {xz2, xzn2}) and two elements (z and z1) whose match in this respect (zn+1, and zn1 respectively) is not a neighbour of 1, it must be the case that {z, z1} and {x, x1, xz, xzn+1, xz2, xzn2} are fixed setwise by '. Repeating this argument shows that '(zi) 2 hzi for every i. Thus, '(hzi) = hzi. Lemma 5.3. Let A = hz1, z2i where |z1| = 2n 6, |z2| = 2, and z1z2 = z2z1, so A ⇠= Z2n ⇥ Z2. Then {z1, z21 } is a suitable pair for D = Dic(A, x2). Also, if = Cay(D, {z±11 , z2, x±1, (xz1)±1, (xz 2 1 ) ±1) then A is invariant under Aut()1. Proof. Checking the conditions for {z1, z21 } to be a suitable pair is straightforward. Since z2 2 S is central in D and x1 = xz2, the following pairs of neighbours of 1 are adjacent in : {x, x1}; {xz1, x1z1}; {xz21 , x1z 2 1 }. However, z1, z 1 1 and z2 have no neighbours in S. Thus, we can distinguish the neighbours of 1 that lie in A from the neighbours of 1 that lie in xA. Repeating this argument shows that every element of A is invariant under Aut()1. Lemma 5.4. Let = Cay(A,S) be an MRR of the abelian group A of Cayley index 2. Let D = Dic(A, y) be a generalised dicyclic group with suitable pair {a1, a2}. Let = Cay(D,S [ {x, x1, xa1, x1a1, xa2, x1a2}) (where x is as in Definition 1.3) and suppose that for every ' 2 Aut()1, we have '(A) = A. If ' is not the identity automorphism, then '(a) = a, and '(xa) = (xa)1 for every a 2 A. Proof. Since '(A) = A and the induced subgraph on A is which has Cayley index 2, we know that we either have '(a) = a for every a 2 A, or '(a) = a1 for every a 2 A. (This is always the case in a Cayley graph of Cayley index 2 on an abelian group.) Similarly, since '(A) = A we have '(xA) = xA. Observe that the induced subgraph on xA is isomorphic to , which has Cayley index 2. This means that there are exactly two graph automorphisms that fix xA and take x to any given vertex xa where a 2 A. Clearly one of these automorphisms is given by left-multiplication by a1, and therefore maps each vertex of the form xa0 to the vertex a1xa0 = xaa0. The other graph automorphism that fixes x and xA (aside from the identity) is the automorphism that maps every vertex xa0 to the vertex x(a0)1. This implies that the other automorphism that maps x to xa must take each vertex of the form xa0 to the vertex a1x(a0)1 = xa(a0)1. In the remainder of this proof, we use NX(v) to denote the set of neighbours of the vertex v that lie in the subset X of the vertices of . First we will show that '(x) 2 {x, x1}. We are assuming that '(A) = A, and need to show that '(x) 62 {xa1, x1a1, xa2, x 1 a2}. Suppose that '(x) /2 {x, x1}. By symmetry, without loss of generality we may assume that '(x) = xa1. Since '(x) = xa1 and '(xA) = xA, as noted above we must have either '(xa) = xaa1 for every a 2 A, or '(xa) = xa1a1 for every a 2 A. Suppose the first of these possibilities holds, so '(xa1) = xa21, which must therefore be a neighbour of 1 in xA, and hence an element of NxA(1) = {x, x1, xa1, x1a1, xa2, x1a2}. J. Morris and J. Tymburski: Most rigid representations and Cayley index 9 Each of these possibilities contradicts one of the properties of being a suitable pair: any of the first four would contradict (ii); either of the last two contradict (iii). If on the other hand the second possibility holds, then '(xa2) = xa12 a1 2 NxA(1). Again, each possible equality contradicts one of the properties of being a suitable pair: either of the first two contradict (i); the third or fourth each contradicts (ii); and either of the last two contradict (iii). We therefore conclude that '(x) 2 {x, x1}, as claimed. Next we show that '(a) = a for every a 2 A. Observe that NA(x 1) = NA(x) = {1, y, a1, ya1, a2, ya2}. Thus, since '(x) 2 {x, x1}, we have '(NA(x)) = NA(x). If '(a) = a1 for every a 2 A, then this implies that a11 2 NA(x), leading to a contradiction to the definition of a suitable pair, as above. (If a11 is any of the first four elements, this contradicts (ii); if it is either of the last two, this contradicts (iv).) Thus, we must have '(a) = a for every a 2 A. Next we show that if '(x) = x then ' = 1. Again as noted above, we must either have '(xa) = xa1 for every a 2 A, or '(xa) = xa for every a 2 A. In the latter case, ' = 1 and we are done. In the former case, we must have '(NA(xa11 )) = NA(xa1). Observe that a1 = xa 1 1 x 1 2 NA(xa11 ), so this would imply that a1 = '(a1) 2 NA(xa1) = {a11 , ya 1 1 , 1, y, a 1 1 a2, ya 1 1 a2}. Similar to the arguments above, each of these possibilities contradicts some property of suitable pairs. If a1 were any of the first four elements of NA(xa1) this would contradict (i); if it were either of the last two, this would contradict (iii). Finally, we show that if '(x) = x1 then '(xa) = (xa)1 for every a 2 A. Again as noted above, we must either have '(xa) = x1a = (xa)1 for every a 2 A, or '(xa) = x1a1 for every a 2 A. In the former case we are done. In the latter case, we must have '(NA(x1a11 )) = NA(xa1). Observe that a1 = x1a 1 1 x 2 NA(x1a 1 1 ), so this would imply that a1 = '(a1) 2 NA(xa1), yielding the same contradiction as in the previous paragraph. Proposition 5.5. Let A1 = hz1i be a cyclic group of order 2n 6, and A2 = hz1, z2i with |z2| = 2 and z1z2 = z2z1. Let S1 = {z1, z11 } and S2 = {z1, z 1 1 , z2}, and let D1 = Dic(A1, zn1 ), and D2 = Dic(A2, z2). Then i = Cay(Di, Si [ {x, x1, xz1, xzn+11 , xz 2 1 , xz n2 1 }) for i 2 {1, 2} is connected and has Cayley index 2 when n 6, and 2 is connected and has Cayley index 2 when n 3. Proof. It is easy to see that S1 is the connection set for a Cayley graph on A1 with Cayley index 2. It is slightly less obvious that S2 is the connection set for a Cayley graph on A2 with Cayley index 2, but becomes clear upon noting that each z1-edge lies in a unique 4- cycle, while each z2-edge lies in two 4-cycles. Fix i 2 {1, 2}, and if i = 1, ensure that n 5. By Lemma 5.2 or Lemma 5.3, we know that {z1, z21 } is a suitable pair for Di, and that for any ' 2 Aut(i)1, '(Ai) = Ai. By Lemma 5.4 with S = Si and this suitable pair, we see that there are only two possibilities for ': ' = 1, or '(a) = a and '(xa) = (xa)1 for every a 2 A. Thus, has Cayley index 2. 10 Art Discrete Appl. Math. 1 (2018) #P1.05 Proposition 5.6. Let A be an abelian group of even order that contains an involution y, and let D = Dic(A, y). Suppose that D has a connected MRR with Cayley index 2. Let A 0 = A⇥ Z2. Then D0 = Dic(A0, y) has Cayley index 2. Proof. Observe that D0 ⇠= D ⇥ Z2. The result is now immediate from Corollary 4.5. As an immediate consequence of Proposition 5.5 and Proposition 5.6, we obtain the following. Corollary 5.7. The following generalised dicyclic groups have Cayley index 2: • Dic(A⇥ Zk2 , zn1 ) where A = hz1i ⇠= Z2n, n 6, and k 0; and • Dic(A ⇥ Zk2 , z2) where A = hz1, z2i ⇠= Z2n ⇥ Z2, |z1| = 2n, |z2| = 2, n 3, and k 0. This leads us to the following theorem. Theorem 5.8. Every generalised dicyclic group that is neither abelian nor a hamiltonian 2-group has Cayley index 2, with the following four exceptions, each of which has Cayley index 4: Dic(Z6, 3), Dic(Z8, 4), Dic(Z10, 5), and Dic(Z4 ⇥ Z2, (0, 1)). Proof. All generalised dicyclic groups of order greater than 96 have Cayley index of 2 (see [10]). To deal with the remaining cases, we begin by considering all abelian groups of even order at most 48. For each group, we choose one representative for each automor- phism class of elements of order 2 to be the distinguished element y = x2. By Corollary 5.7, the result holds for every dicyclic group of order at least 24; this deals with every cyclic group of even order at least 12, all of which have a unique element of order 2. Since Z2 produces an abelian dicyclic group and Z4 produces Q8 which is a hamiltonian 2-group, we need only consider the dicyclic groups over the groups Z6, Z8, and Z10. We note that if n is odd, then Z2n⇥Z2 has only one automorphism class of elements of order 2, so that Corollary 5.7 provides an MRR for the unique generalised dicyclic group over any of these groups when n 3, and in fact produces two MRRs when n > 6. Also, if n is even, Z2n ⇥ Z2 has two automorphism classes of elements of order 2 (the element (n, 1) lies in the same class as (0, 1)). Thus Corollary 5.7 produces an MRR for each of the two possible generalised dicyclic groups over these abelian groups whenever n 6, and an MRR for one of them when n 3. When n = 1 there is a unique generalised dicyclic group which is actually abelian; and when n = 2, one of the two generalised dicyclic groups is the hamiltonian 2-group Q8 ⇥ Z2. Thus we need only consider the two groups Dic(Z4 ⇥ Z2, (0, 1)) and Dic(Z8 ⇥ Z2, (4, 0)). The generalised dicyclic group over Z32 is abelian, and the groups Dic(Z4 ⇥ Z22, (2, 0, 0)) ⇠= Q8 ⇥ Z22 and Dic(Z4 ⇥ Z32, (2, 0, 0, 0)) are hamiltonian 2-groups, so these need not be considered. Finally, if a group has the form D ⇥ Z2 for some smaller generalised dicyclic group D with c(D) = 2, then Corollary 4.5 gives c(D ⇥ Z2) = 2, so we do not have to consider these groups either. This eliminates all generalised dicyclic groups over Z6⇥Z22, Z10⇥Z22, and Z12 ⇥ Z22, as well as Dic(Z8 ⇥ Z22, (0, 1, 0)). J. Morris and J. Tymburski: Most rigid representations and Cayley index 11 With all of this in mind, there are 18 generalised dicyclic groups that remain to be considered. We conclude this section and the paper with Table 4, showing the Cayley index and the connection set for an MRR for each of these generalised dicyclic groups. For four of these groups that have the form D ⇥ Z2 for some smaller generalised dicyclic group D, we use Corollary 4.5, but only after showing that c(D) = 2 in a previous line of the table. For these, instead of explicitly giving the connection set for an MRR, we present the group as D ⇥ Z2. This table completes the proof, and its results are straightforward to verify by computer. Table 4: MRRs for generalised dicyclic groups. Cayley Group index Connection set for an MRR Dic(Z6, 3) 4 {z±11 , x±1} Dic(Z8, 4) 4 {z±11 , x±1} Dic(Z10, 5) 4 {z±11 , x±1} Dic(Z4 ⇥ Z2, (0, 1)) 4 {z±11 , x±1, (z1x)±1} Dic(Z8 ⇥ Z2, (4, 0)) 2 {z±11 , z2, x±1, (z1x)±1, (z2x)±1} Dic(Z4 ⇥ Z4, (2, 0)) 2 {z±11 , z±12 , (z1z2)±1, x±1, (z1x)±1} Dic(Z4 ⇥ Z22, (0, 1, 0)) 2 {z±11 , z3, x±1, (z1x)±1, (z3x)±1} Dic(Z6 ⇥ Z3, (3, 0)) 2 {z±11 , z±12 , x±1, (z2x)±1, (z1z2x)±1} Dic(Z8 ⇥ Z22, (4, 0, 0)) 2 D ⇠= Dic(Z8 ⇥ Z2, (4, 0))⇥ Z2 Dic(Z8 ⇥ Z4, (4, 0)) 2 {z±11 , z±12 , x±1, (z61z12 x)±1, (z51z2x)±1} Dic(Z8 ⇥ Z4, (0, 2)) 2 {z±11 , z±12 , x±1, (z51x)±1, (z31z2x)±1} Dic(Z24 ⇥ Z2, (2, 0, 0)) 2 D ⇠= Dic(Z4 ⇥ Z4, (2, 0))⇥ Z2 Dic(Z24 ⇥ Z2, (0, 0, 1)) 2 {z±11 , z±12 , x±1, (z32x)±1, (z31z22x)±1} Dic(Z4 ⇥ Z32, (0, 1, 0, 0)) 2 D ⇠= Dic(Z4 ⇥ Z22, (0, 1, 0))⇥ Z2 Dic(Z12 ⇥ Z3, (6, 0)) 2 {z±11 , z±12 , x±1, (z71z2x)±1, (z31z2x)±1} Dic(Z6 ⇥ Z6, (3, 0)) 2 D ⇠= Dic(Z6 ⇥ Z3, (3, 0))⇥ Z2 Dic(Z12 ⇥ Z4, (6, 0)) 2 {z±11 , z±12 , x±1, (z41z2x)±1, (z91z32x)±1} Dic(Z12 ⇥ Z4, (0, 2)) 2 {z±11 , (z31z2)±1, x±1, (z1x)±1, (z1z2x)±1} References [1] D. P. Byrne, M. J. Donner and T. Q. Sibley, Groups of graphs of groups, Beitr. Algebra Geom. 54 (2013), 323–332, doi:10.1007/s13366-012-0093-7. [2] C.-y. Chao, On a theorem of Sabidussi, Proc. Amer. Math. Soc. 15 (1964), 291–292, doi:10. 2307/2034055. [3] C. D. Godsil, GRRs for nonsolvable groups, in: L. Lovász and V. T. Sós (eds.), Algebraic Methods in Graph Theory, Volume I, North-Holland, Amsterdam, volume 25 of Colloquia Mathematica Societatis János Bolyai, pp. 221–239, 1981, papers from the conference held in Szeged, August 24 – 31, 1978. 12 Art Discrete Appl. Math. 1 (2018) #P1.05 [4] M. Hall, Jr., The Theory of Groups, Macmillan, New York, 1959. [5] D. Hetzel, Über reguläre graphische Darstellung von auflösbaren Gruppen, Diploma thesis, Technische Universität Berlin, Berlin, 1976. [6] W. Imrich, Graphen mit transitiver Automorphismengruppe, Monatsh. Math. 73 (1969), 341– 347, doi:10.1007/bf01298984. [7] W. Imrich, Graphs with transitive Abelian automorphism group, in: P. Erdős and A. Rényi (eds.), Combinatorial Theory and its Applications, Volume II, North-Holland, Amsterdam, vol- ume 4 of Colloquia Mathematica Societatis János Bolyai, pp. 651–656, 1970, proceedings of the Colloquium on Combinatorial Theory and its Applications held at Balatonfüred, August 24 – 29, 1969. [8] W. Imrich, On graphical regular representations of groups, in: A. Hajnal, R. Rado and V. T. Sós (eds.), Infinite and Finite Sets, Volume II, North-Holland, Amsterdam, volume 10 of Colloquia Mathematica Societatis János Bolyai, pp. 905–925, 1975, proceedings of a Colloquium held at Keszthely, June 25 – July 1, 1973 (dedicated to Paul Erdős on his 60th birthday). [9] W. Imrich, On graphs with regular groups, J. Comb. Theory Ser. B 19 (1975), 174–180, doi: 10.1016/0095-8956(75)90082-9. [10] W. Imrich and M. E. Watkins, On automorphism groups of Cayley graphs, Period. Math. Hun- gar. 7 (1976), 243–258, doi:10.1007/bf02017943. [11] L. A. Nowitz and M. E. Watkins, Graphical regular representations of non-abelian groups, I, Canad. J. Math. 24 (1972), 993–1008, doi:10.4153/cjm-1972-101-5. [12] L. A. Nowitz and M. E. Watkins, Graphical regular representations of non-abelian groups, II, Canad. J. Math. 24 (1972), 1009–1018, doi:10.4153/cjm-1972-102-3. [13] G. Sabidussi, Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426–438, doi:10.1007/ bf01304186. [14] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.6, 2016, http: //www.gap-system.org/. [15] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.5.1), 2017, http://www.sagemath.org/. [16] M. E. Watkins, On the action of non-Abelian groups on graphs, J. Comb. Theory Ser. B 11 (1971), 95–104, doi:10.1016/0095-8956(71)90019-0. [17] M. E. Watkins, On graphical regular representations of Cn ⇥ Q, in: Y. Alavi, D. R. Lick and A. T. White (eds.), Graph Theory and Applications, Springer, Berlin, volume 303 of Lecture Notes in Mathematics, pp. 305–311, 1972, proceedings of the Conference at Western Michigan University, Kalamazoo, Michigan, May 10 – 13, 1972 (dedicated to the memory of J. W. T. Youngs). [18] M. E. Watkins, Graphical regular representations of alternating, symmetric, and miscellaneous small groups, Aequationes Math. 11 (1974), 40–50, doi:10.1007/bf01837731. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.06 https://doi.org/10.26493/2590-9770.1243.8ac (Also available at http://adam-journal.eu) Fault-Hamiltonicity of Cartesian products of directed cycles ⇤ Chun-Nan Hung Department of Information Management, Da-Yeh University, Changhua, 51591, Taiwan, R.O.C. Tao-Ming Wang Department of Applied Mathematics, Tunghai University, Taichung, 40704, Taiwan Lih-Hsing Hsu Department of Computer Science and Information Engineering, Providence University, Taichung, 43301, Taiwan, R.O.C. Eddie Cheng Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA Received 19 July 2017, accepted 13 August 2017, published online 13 February 2018 Abstract Although the Cartesian product of two Hamiltonian graphs is Hamiltonian, the corre- sponding statement for directed graphs is not true. Indeed, it is known that it does not always hold even for the Cartesian products of two directed cycles. In this paper, we study the Cartesian product and its generalization of a directed graph G and a directed cycle. We show that if G has “strong” fault-Hamiltonicity properties, then so does G⇤Cn, that is, the Cartesian product of G and a cycle of length n. We also discuss some related problems. Keywords: Digraphs, fault-Hamiltonicity, Cartesian product. Math. Subj. Class.: 05C20, 05C45 ⇤We would like to thank the anonymous referees for a number of helpful comments and suggestions including identifying a couple of gaps in the proof of the main result in the original manuscript. E-mail addresses: spring@mail.dyu.edu.tw (Chun-Nan Hung), wang@thu.edu.tw, wang@go.thu.edu.tw (Tao-Ming Wang), lhhsu@pu.edu.tw (Lih-Hsing Hsu), echeng@oakland.edu (Eddie Cheng) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.06 1 Introduction The interconnection network is one of the center pieces of a parallel architecture. The underlying topology of such a parallel machine is a graph, usually referred to as an in- terconnection network. Depending on the applications, the graph may be undirected or directed. A Hamiltonian cycle in a graph is a cycle that visits every vertex of the graph (exactly once). (If the underlying graph is directed, then a cycle means a directed cycle.) A graph is Hamiltonian if it has a Hamiltonian cycle. A Hamiltonian path from u to v in a graph is a path from u to v that visits every vertex of the graph. (Again, if the underlying graph is directed, then a path means a directed path.) A graph is Hamiltonian connected if there exists a Hamiltonian path from u to v for every distinct ordered pair of vertices u and v. Hamiltonicity is an important issue in the study of interconnection networks and there are many papers in this area. Paper [8] contains many references in this area and we refer the readers to [8] for an extensive list of references on Hamiltonicity related problems in interconnection networks. (A small partial list of such papers is [3, 4, 5, 7, 10, 13, 14].) However, most research has been done in the undirected setting as the analysis is, in gen- eral, more complicated in the directed case. A directed graph G is k-regular if the in-degree and out-degree of every vertex is k. So a connected 1-regular directed graph is a directed cycle. (We will simply refer to directed cycles as cycles if it is clear from the context.) The Cartesian product of two directed graphs G1 and G2 is the directed graph G1 ⇤G2 = (V,E) where V = V1 ⇥ V2 and ((u1, v1), (u2, v2)) is in E if either (1) u1 = u2 and (v1, v2) 2 E2, or (2) v1 = v2 and (u1, u2) 2 E1. The Cartesian product of undirected graphs can be defined similarly. (One can check that given three directed graphs G1, G2 and G3, (G1 ⇤G2)⇤G3 is isomorphic to G1 ⇤ (G2 ⇤G3). Thus the Cartesian product of finitely many directed graphs can be nat- urally defined.) Cartesian product is an important topic in the study of interconnection networks. For example, the classical hypercube is Kn2 , that is, a Cartesian product of n complete graphs on two vertices. Although the Cartesian product of two Hamiltonian graphs is always Hamiltonian, this is false for directed graphs. Trotter and Erdős [12] gave a necessary and sufficient condition for the Cartesian product of two Hamiltonian directed cycles to be Hamiltonian. To be precise, let gcd(m,n) denote the greatest common divisor of two positive integers m and n. Then the Cartesian product of two directed cycles Cm and Cn is Hamiltonian if and only if gcd(m,n) 2 and there exists positive integers d1 and d2 such that gcd(m,n) = d1 + d2, gcd(m, d1) = 1, and gcd(n, d2) = 1. So C2 ⇤C3 is not Hamiltonian since gcd(2, 3) = 1 < 2. Vertices in an interconnection network represent processors and edges represent links between processors. Since processors and links may fail, it is meaningful to study such faulty networks. A graph G = (V,E) is k-Hamiltonian if G F is Hamiltonian for every F ✓ V [ E and |F |  k. Similarly, a graph G = (V,E) is k-Hamiltonian-connected if G F is Hamiltonian-connected for every F ✓ V [ E and |F |  k. Here F is the set of faults that represent failed processors (vertices) and failed links (edges). We note that if G = (V,E) is k-Hamiltonian-connected, then G is k-Hamiltonian whenever |V | > k + 2. For undirected graphs, many related results on k-Hamiltonicity and k-Hamiltonian connectedness with respect to the Cartesian product are known. See, for example, [1, 6, 11]. We have already mentioned the interesting result given in [12]. It is even more inter- esting if one considers the Cartesian product of three directed cycles. In particular, one can C.-N. Hung et al.: Fault-Hamiltonicity of Cartesian products of directed cycles 3 check by brute force that C2 ⇤C3 ⇤C4 is a 3-regular, 2-Hamiltonian and 1-Hamiltonian- connected directed graph. In fact, C2 ⇤C3 ⇤C5, C2 ⇤C3 ⇤C6, C2 ⇤C4 ⇤C5 and C2 ⇤C5 ⇤C5 are also 3-regular, 2-Hamiltonian and 1-Hamiltonian-connected directed graphs. Results similar to the one given in [12] appeared in [2, 9]. This gives an indication that Hamiltonicity problems for directed graphs are more difficult than the undirected ver- sion. In addition, it is proved in [2] that every product of more than two directed cycles is Hamiltonian. The ultimate goal is to obtain a result on k-Hamiltonicity and k-Hamiltonian connect- edness with respect to the Cartesian product of directed graphs. Given the above exam- ple, we believe that this problem is difficult. Thus we study directed graphs of the form G⇤Cn. We want to show that if G has “strong” Hamiltonicity property, then so does G⇤Cn. In fact, we will generalize the concept of Cartesian product by considering the following. Let G be a set of directed graphs, each with the same fixed number of ver- tices. We say that G has a certain property if every directed graph in G has this prop- erty. Now we take n graphs G0, G1, . . . , Gn1 from G with repetitions allowed. Let fi : V (Gi) ! V (Gi+1), for i = 0, 1, . . . , n 1, be bijections where addition is taken modulo n. We construct the directed graph H = (V,E) by letting V = [n1i=0 V (Gi) and E = [n1i=0 E(Gi) [ [n1i=0 {(u, fi(u)) : u 2 V (Gi)} . We call H an n-G-directed graph. So G⇤Cn is an n-{G}-directed graph. For notational simplicity, we denote ECi = {(u, fi(u)) : u 2 V (Gi)} and we let CGij be the subgraph of H induced by [jr=iV (Gr) (modulo n). Given (u, v) 2 ECi, we may refer to v as fi(u) and u = f 1 i (v). For the case G⇤Cn, we may simply refer to fi as f . Whenever we refer to a range [i, j], it is considered modulo n. In this paper, we consider deleting vertices and arcs. As mentioned before, these deleted elements correspond to failed processors and links in an interconnection network, and we refer them as faults. Let G be an r-regular directed graph. Clearly the best one can hope for is for G to be (r 1)-Hamiltonian and (r 2)-Hamiltonian connected. As pointed out earlier, there exist directed graphs that achieve such optimal properties when r = 3. In this paper, we show that if G has such optimal properties, then so does G⇤Cn. In fact, our result covers the more general n-G-directed graph. At first glance, one may wonder whether this is consistent with the necessary and sufficient condition given by Trotter and Erdős for Cn ⇤Cm to be Hamiltonian. After all, Cm is 1-regular and Hamiltonian but Cm ⇤Cn may not be Hamiltonian. One may argue that in this case, the condition “1”-Hamiltonian connected is meaningless. As we shall see, our main result requires the regularity of G to be at least 3. 2 The main result In this section, we present our main result. We want to show that if G has good Hamiltonian properties, then so does an n-G-directed graph. We start with the following lemma. Lemma 2.1. Let k 2 and N k + 5. Let G be a class of (k + 1)-regular and (k 1)- Hamiltonian-connected graphs on N vertices. Let H be an n-G-directed graph obtained from G0, G1, . . . , Gn1 in G with the corresponding bijections f0, f1, . . . , fn1. Let [i, j] be a range. Let Fr ✓ V (Gr) [ E(Gr) for every r in the range [i, j]. Let Fr,r+1 ✓ ECr. Let s and t be vertices in Gi Fi and Gj Fj , respectively. Suppose 1. |Fr|  k 1 for every r in the range [i, j] and 4 Art Discrete Appl. Math. 1 (2018) #P1.06 2. |Fr|+ |Fr+1|+ |Fr,r+1|  k + 2 for every r in the range [i, j 1]. Then there is a Hamiltonian path from s to t in CGi,j ([jr=iFr) ([ j1 r=iFr,r+1). Proof. If i = j, then there is nothing to prove as Gi is (k1)-Hamiltonian-connected. For notational simplicity, we may assume that i = 1. We consider two cases. Case 1: j = 2. We want to find an arc (u1, v2) 2 EC1 F1,2 where u1 2 V (G1) (F1 [ {s}) and v2 2 V (G2) (F2 [ {f1(s)}) and (u1, v2) 6= (f11 (t), t). Such an arc exists if N > |F1|+ |F2|+ |F1,2|+ |{(s, f1(s))}|+ |{(f11 (t), t)}|. But |F1| + |F2| + |F1,2|  k + 2. Thus we are done as N > k + 2 + 2 = k + 4. We now obtain a desired Hamiltonian path by using a Hamiltonian path from s to u1, the arc (u1, v2) and a Hamiltonian path from v2 to t. Case 2: j 3. We first find an arc (u1, v2) 2 EC1 F1,2 where u1 2 V (G1) (F1 [ {s}) and v2 2 V (G2) (F2 [ {f1(s)}). Such an arc exists if N > |F1|+ |F2|+ |F1,2|+ |{(s, f1(s)}|. But |F1|+ |F2|+ |F1,2|  k + 2. Thus we are done as N > k + 2 + 1 = k + 3. Similarly, we can obtain an arc (u2, v3) 2 EC2 F2,3 where u2 6= v2, and so on, via an inductive argument, in obtaining (ui, vi+1)’s, until we obtain an arc (uj2, vj1) 2 ECj2 Fj2,j1 where uj2 2 V (Gj2) (Fj2 [ {vj2}). Now, we need to find an arc (uj1, vj) 2 ECj1 Fj1,j where uj1 2 V (Gj1) (Fj1 [ {vj1}) and vj 2 V (Gj) (Fj [ {t}) which can be guaranteed since N > |Fj1|+ |Fj |+ |Fj1,j |+ 2 (as |Fj1|+ |Fj |+ |Fj1,j |  k + 2 and N k+5). Now since Gr is (k 1)-Hamiltonian-connected, we have a Hamiltonian path from vr to ur in Gr for every r in [i, j] with v1 = s and uj = t. These paths together with the arcs (ur, vr+1)’s give a desired Hamiltonian path. We remark that if we replace (2) by |Fr|+ |Fr+1|+ |Fr,r+1|  k+1 for every r in the range [i, j 1] in Lemma 2.1, then the assumption that N k + 5 can be replaced with the weaker assumption that N k + 4. Theorem 2.2. Let k 2 and n 3. Let G be a class of (k + 1)-regular, k-Hamiltonian and (k 1)-Hamiltonian connected graphs on N vertices. Let H be an n-G-directed graph. Then H is (k + 2)-regular. Moreover H is (k + 1)-Hamiltonian if N k + 4 and k-Hamiltonian connected if N k + 5 and k 3. Proof. We first prove that H is (k + 1)-Hamiltonian. Let F be a set of faults with |F |  k + 1. We let Fi be the set of faults in Gi. We consider two cases. Case 1: |Fi| = k+1 for some i. Without loss of generality, we may assume that |F0| = k+1. Let x 2 F0 and define F 00 = F0 {x}. By assumption, there is a Hamiltonian cycle C 0 0 in G0F 00. Regardless of whether x is a vertex or an arc, C 00{x} is a Hamiltonian path P 0 0 from u to v for some u and v. Now let y = f0(v) and z = f 1 n1(u). By Lemma 2.1, there is a Hamiltonian path from y to z in the CG1,n1 ([n1r=1Fr) = CG1,n1. (Note C.-N. Hung et al.: Fault-Hamiltonicity of Cartesian products of directed cycles 5 that equality holds as Fr = ; for r 2 {1, 2, . . . , n 1}.) This together with P 00 gives a Hamiltonian cycle in H F . Case 2: |Fi|  k for every i. We first note that 2k > k + 1 as k 2. Thus there is at most one i with |Fi| = k. Therefore we may assume that |F0| is the largest and |Fi|  k1 for i 6= 0. Now, by assumption, there is a Hamiltonian cycle C0 in G0 F0. We want to find an arc (v, u) in C0 such that (v, f0(v)), (f 1 n1(u), u), f0(v), f 1 n1(u) 62 F. Here |Fr|+ |Fr+1|+ |Fr,r+1|  k+ 1 for r 2 {0, 1, 2, . . . , n 2} as |F |  k+ 1. So we only require N k + 4 from the remark after Lemma 2.1. Now C0 has at least N |F0| arcs. Since N |F0| > |F | |F0|, such (v, u) exists. Now the argument in Case 1 applies, and we are done. This completes the proof for H being (k + 1)-Hamiltonian. The case for H being k- Hamiltonian connected is much more difficult. We assume N k + 5. (We will see later why k+5 is needed.) Let F be a set of faults with |F |  k and we define the Fi’s as before. Let s and t be two fault-free vertices and our goal is to construct a Hamiltonian path from s to t in H F . We consider two main cases. (Unfortunately, subcases are needed here.) Case 1: |Fi| = k for some i. Without loss of generality, we may assume that |F0| = k. So all the faults are in F0. We have to consider subcases depending on the locations of s and t. Subcase 1.1: s and t are in G0 F0. Let x 2 F0 and define F 00 = F0 {x}. By assumption, there is a Hamiltonian path P 00 from s to t in G0 F 00. Regardless of whether x is a vertex or an arc, P 00{x} contains the following two disjoint paths that span G0F0: Q0 from s to u and Q00 from v to t for some u and v. (It is possible that s = u or v = t.) Moreover, Q0 and Q00 cover all the vertices in G0 F0. Now let y = f0(u) and z = f1n1(v). By Lemma 2.1, there is a Hamiltonian path from y to z in CG1,n1 ([n1r=1Fr) = CG1,n1. This, together with the edges (u, y) and (z, v), Q0 and Q00, gives a Hamiltonian path from s to t in H F . Subcase 1.2: s is in G0F0 and t is in GiFi = Gi where i 6= 0. If i = n1, then it is straightforward as G0 F0 is a Hamiltonian. (Since G0 F0 is Hamiltonian, there is a Hamiltonian path Q0 from s to y in G0 F0 for some y. Now apply Lemma 2.1 to obtain a Hamiltonian path from f0(y) to t in CG1,n1 ([n1r=1Fr) = CG1,n1. This, together with the edge (y, f0(y)) and Q0, gives a Hamiltonian path from s to t in H F .) Thus we may assume that i 6= n 1. By assumption, G0 F0 has a Hamiltonian cycle C0. Since N k + 5, there exists a vertex u0 on C0 such that (u0, s) is not an arc in C0 and ui 6= t where u1 = f0(u0), u2 = f1(u1), . . . , ui = fi1(ui1). (It is possible that u0 = s.) Now C0 contains the following two disjoint paths that span G0 F0: Q0 from s to u0 and Q00 from v to x for some v and x as determined by C0 and Q0. (It is possible that v = x.) We note that since (u0, s) is not an arc, Q00 is not empty. Now, apply Lemma 2.1 to obtain a Hamiltonian path P1 from fi(ui) to f1n1(v) in CGi+1,n1 ([ n1 r=i+1Fr) = CGi+1,n1. We apply Lemma 2.1 again, this time to obtain a Hamiltonian path P2 from f0(x) to t in CG1,i ([ir=1(Fr [ {ur})) = CG1,i ([ir=1{ur}). For the moment, assume that f0(x) 6= t. Then Q0, (u0, u1, . . . , ui, fi(ui)), P1, (f 1 n1(v), v), Q 0 0, (x, f0(x)), P2 is a desired Hamiltonian path from s to t in H F . (See Figure 1.) 6 Art Discrete Appl. Math. 1 (2018) #P1.06 t s u0 v x u1 Q0 f (x)0 P2 u2 uiP1 f -1 (v)n-1 u = f (u )ii + 1 i Q0‘ Figure 1: The Hamiltonian path of Subcase 1.2. C.-N. Hung et al.: Fault-Hamiltonicity of Cartesian products of directed cycles 7 The remaining possibility is f0(x) = t. Then i = 1. This case is actually simpler as we can obtain a desired Hamiltonian path by using Q0, a Hamiltonian path from u1 to f 1 n1(v) in CG1,n1 {t} (via Lemma 2.1), the edge (f 1 n1(v), v), the path Q00, and the edge (x, t). Subcase 1.3: t is in G0 F0 and s is in Gi Fi where i 6= 0. This is similar to Sub- case 1.2 by observing instead of going from G0 to Gi via G1, G2, . . . , Gi1 to obtain a di- rected path from s to t, we can “trace backward” from to t to s via Gn1, Gn2, . . . , Gi+1. To be precise, we let GR be the directed graph obtained from G F by reversing the di- rection on every arc. Then a directed path from s to t in G F can be obtained from a directed path from t to s in GR, whose existence is proved in Subcase 1.2. Subcase 1.4: s and t are in GiFi where i 6= 0. We have to consider several scenarios. We first assume that i = n 1. By assumption, there is a Hamiltonian path P from s to t in Gn1 Fn1 = Gn1. Choose any (u, v) on P such that fn1(u) 62 F0 = F . (Again such u exists as N k + 5. Henceforth, we will not explicitly mention this when choosing an appropriate vertex.) Now P contains the following two disjoint paths that span Gn1 Fn1 = Gn1: Q from s to u and Q0 from v to t. (It is possible that s = u or v = t.) By assumption, there is a Hamiltonian cycle in G0F0, which implies that there is a Hamiltonian path P from fn1(u) to w in G0F0 for some w. Now apply Lemma 2.1 to obtain a Hamiltonian path R from f0(w) to f1n2(v) in CG1,n2([ n2 r=1Fr) = CG1,n2. Now Q, (u, fn1(u)), P, (w, f0(w)), R, (f 1 n2(v), v), Q 0 is a desired Hamiltonian path from s to t in H F . We now assume i = 1. By assumption, there is a Hamiltonian cycle in G0 F0, which implies that there is a Hamiltonian path P from u to v in G0 F0 for some u and v. We may choose v such that f0(v) 62 {s, t}. By assumption, there is a Hamiltonian path Q from f0(v) to t in G1 (F1 [ {s}) = G1 {s}. Now by Lemma 2.1, there is a Hamiltonian path R from f1(s) to f1n1(u) in CG2,n1 ([ n1 r=2Fr) = CG2,n1. Now (s, f1(s)), R, (f 1 n1(u), u), P, (v, f0(v)), Q is a desired Hamiltonian path from s to t in H F . We may now assume that 2  i  n 2. By assumption, there is a Hamiltonian cycle in G0 F0, which implies that there is a Hamiltonian path P0 from u to v in G0 F0 for some u and v. By assumption, there is a Hamiltonian path Pi from s to t in Gi Fi = Gi. Pick any (y, z) on Pi such that fi(y) 6= f1n1(u) and f 1 i1(z) 6= f0(v). Now Pi contains two disjoint paths that span Gi Fi = Gi: Qi from s to y and Q0i from z to t. Apply Lemma 2.1 to get a Hamiltonian path R from f0(v) to f1i1(z) in CG1,i1 ([ i1 r=1Fr) = CG1,i1. Apply Lemma 2.1 to get a Hamiltonian path R0 from fi(y) to f1n1(u) in CGi+1,n1 ([n1r=i+1Fr) = CGi+1,n1. Now Qi, (y, fi(y)), R 0 , (f1n1(u), u), P0, (v, f0(v)), R, (f 1 i1(z), z), Q 0 i is a desired Hamiltonian path from s to t in H F . (See Figure 2.) Subcase 1.5: s is in GiFi and t is in Gj Fj where 1  i < j  n 1. By assump- tion, there is a Hamiltonian cycle in G0 F0, which implies that there is a Hamiltonian path P0 from u to v in G0 F0 for some u and v. We first assume that i 6= 1. We may assume that f0(v) 6= s. By Lemma 2.1, we obtain a Hamiltonian path P 0 from f0(v) to w 8 Art Discrete Appl. Math. 1 (2018) #P1.06 R R‘ u vf -1 (u)n-1 f (v)0 f -1 (z)i - 1 f (y)i Qi Qi‘ s y z t P0 Figure 2: The Hamiltonian path of Subcase 1.4. C.-N. Hung et al.: Fault-Hamiltonicity of Cartesian products of directed cycles 9 in CG1,i1 for some w in Gi1 Fi1 = Gi1 to be determined. By assumption, there is a Hamiltonian path Pi from s to y in Gi Fi = Gi for some y such that y 6= s and fi(y) 6= t. Let choose (x, z) on Pi such that fj1(· · · (fi+1(fi(x))) 6= t and f1i1(z) 6= f0(v). (If n 1 = j + 1, we further require that fj(fj1(· · · (fi+1(fi(x)))) 6= f1n1(u).) Now we choose w = f1i1(z). Now let A = (x, fi(x), . . . , fj1(· · · (fi+1(fi(x)))). Let y0 = fi(y) if j = i + 1 and y0 be any vertex in Gj {t, fj1(· · · (fi+1(fi(x)))} otherwise. (If j = i + 2, we further require fi(y) 6= f1j1(y0).) For notational convenience, let x 0 = fj1(· · · (fi+1(fi(x))). Let A0 be (y, y0) if j = i + 1 and let A0 be a Hamiltonian path from fi(y) to f1j1(y 0) in CGi+1,j1 ([j1r=i+1{fr1(· · · (fi+1(fi(x)))}). Now Pi contains two disjoint paths that span Gi Fi = Gi: Qi from s to x and Q0i from z = fi1(w) to y. Now let R be a Hamiltonian path from y0 to t in Gj {x0} and R0 be a Hamiltonian path from fj(x0) to f1n1(u) in CGj+1,n1. Then Qi, A, (x 0 , fj(x 0)), R0, (f1n1(u), u), P0, (v, f0(v)), P 0 , (w, z), Q0i, (y, fi(y)), A 0 , (f1j1(y 0), y0), R is a desired Hamiltonian path from s to t in H F . (See Figure 3.) If i = 1, then a small adjustment is needed for P 0. We note that there is freedom in choosing v, y and z. However, once v is chosen, u is forced; similarly, once z is chosen, x is forced. Here we reduce the degree of freedom by one and choose v such that z = f0(v) in addition to the other restrictions. It is not difficult to see that such v can be chosen. So P 0 is the arc (v, f0(v)). Subcase 1.6: s is in GjFj and t is in GiFi where 1  i < j  n1. We construct a desired Hamiltonian path in several steps. (This is similar to Subcase 1.5 but it is slightly more complicated.) By assumption, there is a Hamiltonian cycle C0 in G0 F0. We want to find two arcs (v, u) and (x, y) on C0 to delete so that C0 contains two disjoint paths that span G0 F0: Q0 from u to x and Q00 from y to v. It is possible that u = x or y = v. (But both cannot occur at the same time.) There are only a few restrictions on the candidacies of (v, u) and (x, y). We call the path (v, f0(v), f1(f0(v)), . . . , fi1(· · · f1(f0(v))), fi(fi1(· · · f1(f0(v)))), R1; and the requirement is fi1(· · · f1(f0(v))) 6= t. For convenience, let w = fi(fi1(· · · f1(f0(v)))). (For the case i + 1 = j, then w is in Gj and we further require w 6= s.) We will call the path (f1j (· · · f 1 n2(f 1 n1(u))), . . . , f 1 n2(f 1 n1(u)), f 1 n1(u), u), R2; moreover, the requirement is f1j (· · · f 1 n2(f 1 n1(u))) 6= s. For convenience, let w 0 = f1j (· · · f 1 n2(f 1 n1(u))). (For the case i + 1 = j, then w is in Gj and we further require w0 6= w.) Now let R3 be a Hamiltonian path from w to w0 in CGi+1,j{s}. It turns out that the case j = n 1 and fn1(s) 2 F requires modification of our construction. So for now, we assume that this is not the case. (Note that j = n 1 and fn1(s) 2 F implies (s, fn1(s)) is fault-free as F = F0.) We have more freedom for (x, y) in most 10 Art Discrete Appl. Math. 1 (2018) #P1.06 u v R‘ R A G j P‘ f -1 (u)n-1 ‘x f ‘(x )j f (x)i A‘ t ‘y f -1 (y )j -1 ‘ f (y)i y s x Qi z w = f -1 (z)j - 1 f (v)0P 0 Qi‘ Gi Figure 3: The Hamiltonian path of Subcase 1.5. C.-N. Hung et al.: Fault-Hamiltonicity of Cartesian products of directed cycles 11 instances. If i = 1, we require f0(x) 6= t. If j = n 1 and fn1(s) 62 F , then we let y = fn1(s). (Recall that the case j = n 1 and fn1(s) 2 F is deferred.) We further note that if j = n 1, there is only one choice for y and hence there is only one choice of (x, y), so we should pick (x, y) first and then (v, u). If j 6= n 1, let A be the path consisting of (s, fj(s)), the Hamiltonian path from fj(s) to f1n1(y) in CGj+1,n1 [n1r=j+1{fr1(fr2(· · · fj(w)))} and (f 1 n1(y), y); otherwise (that is, j = n 1 and fn1(s) 62 F ), let A = (s, y). (Note that if j 6= n 1, then (s, fj(s)) is fault- free as F = F0.) Let B be the path consisting of (x, f0(x)) and the Hamiltonian path from f0(x) to t in CG1,i [ir=1{fr1(fr2(· · · f0(v)))}. Then A,Q 0 0, R1, R3, R2, Q0, B is a desired Hamiltonian path from s to t in H F . (See Figure 4.) Now we con- sider the case when j = n 1 and fn1(s) 2 F . Then we consider the k + 1 arcs (s, s1), (s, s2), . . . , (s, sk+1) in Gj that start at s. Since |F | = k, we may assume, without loss of generality, that fn1(s1) 62 F . So we let y = fn1(s1), and the path A will be (s, s1, y). So we pick (x, y) first, then we pick (v, u) as before but we now need to include the restriction that w0 6= s1. We note that then R3 needs to be a Hamiltonian path from w to w0 in CGi+1,n1 {s, s1}. Since k 3, such a path exists via the usual explanation. We remark that one can actually adjust the proof to give a construction for k = 2. Case 2: |Fi|  k 1 for every i. We consider two subcases depending on the locations of s and t. Subcase 2.1: s and t belong to the same Gi Fi. Without loss of generality, we may assume that s and t belong to G0 F0. By assumption, there is a Hamiltonian path P0 from s to t in G0 F0. Choose (u, v) on P0 such that f0(u), (u, f0(u)) 62 F and f 1 n1(v), (f 1 n1(v), v) 62 F . Now apply Lemma 2.1 to obtain a Hamiltonian path from f0(u) to f1n1(v) in CG1,n1 ([ n1 r=1Fr). Now, the usual argument gives a desired Hamiltonian path. Subcase 2.2: s and t belong to different Gi Fi’s. We may assume that s belong to G0 F0 and t belong to Gj Fj where j 6= 0. If j = n 1, then this result follows directly from Lemma 2.1. So we may assume that j  n 2. Subsubcase 2.2.1: |Fi|  k 2 for every i. Find x in G0 F0 such that f0(x) 6= t (if j = 1). We remark that the argument in this subcase requires only |F1|, |F2|, . . . , |Fj |  k 2. By assumption, there is a Hamiltonian path P0 from s to x in G0 F0. We want to find (u, v) on P0 to delete so that P0 contains two disjoint paths that span G0 F0: Q0 from s to u and Q00 from v to x. There are only a few restrictions on the candidacy of (u, v). We want f1n1(v), (f 1 n1(v), v) 62 F , the path (u, f0(u), f1(f0(u)), . . . , y = fj1(fj2(· · · f1(f0(u)))), fj(y)) be a fault-free path, and y 6= t. (We note that there is a definition embedded in the path. The penultimate vertex is fj1(fj2(· · · f1(f0(u)))) which we call y. Thus the last vertex is fj(y).) It is easy to see that such an edge (u, v) exists. Let R1 be (f0(u), f1(f0(u)), . . . , y). Let R2 be the Hamiltonian path from f0(x) to t in CG1,j ([jr=1Fr) {f0(u), f1(f0(u))), . . . , y}. (Such a path exists by Lemma 2.1 since |Fr|  k 2 and we delete at most one additional vertex from each Gr so |Fr|0 + |Fr+10 |+ |Fr,r+1|  k + 2 where F 0r is Fr. Here we need N k + 5.) Let R3 be the Hamiltonian path from fj(y) to f1n1(v) 12 Art Discrete Appl. Math. 1 (2018) #P1.06 R1 R2 R3 A B u v Q0 y x tw s w1 f -1 (u)n-1 f -1 (y)n-1 f (v)0 f (x)0 G G i f (s)j j Q0‘ Figure 4: The Hamiltonian path of Subcase 1.6. C.-N. Hung et al.: Fault-Hamiltonicity of Cartesian products of directed cycles 13 in CGj+1,n1 ([n1r=j Fr). Then Q0, (u, f0(u)), R1, (y, fj(y)), R3, (f 1 n1(v), v), Q 0 0, (x, f0(x)), R2 is a desired Hamiltonian path from s to t in H F . (See Figure 5.) y t s uv R2 R1 Q0 Q0 f -1 (v)n-1 f (y)j f (x)0 f (u)0 ‘ x R3 Figure 5: The Hamiltonian path of Subsubcase 2.2.1. Henceforth, |Fi| = k 1 for some i. Since k 3, such an i is unique. Subsubcase 2.2.2: |F0| = k 1 or |Fq| = k 1 for some q = j + 1, j + 2, . . . , n 1. Then the argument of Subsubcase 2.2.1 applies by the remark given at the start of its argu- ment. Subsubcase 2.2.3: |Fj | = k 1. We will adjust the construction given in Subsub- case 2.2.1. We note that there is at most one fault not in Fj . We find a vertex y 6= t in Gj Fj such that (f10 (· · · f 1 j1(y)), . . . , f 1 j1(y), y) is fault-free. Now for this chosen y, let Pj be a Hamiltonian path from y to t in Gj Fj . We find an arc (w,w0) on Pj such that (w, fj(w)) and (f1j1(w 0), w0) are fault-free. (We allow y = w or w0 = t.) If 14 Art Discrete Appl. Math. 1 (2018) #P1.06 j = 1, we further require f10 (w0) 6= s. Let P y j and P t j be the subpaths of Pj from y to w and from w0 to t, respectively. Let u = f10 (· · · f 1 j1(y)) and R1 be (u, . . . , f 1 j1(y), y), followed by P yj and (w, fj(w)). If j = 1, then let x = f 1 0 (w 0). Otherwise, we pick x in G0 (F0 [ {s, u}) such that (x, f0(x)) is fault-free. (If j = 2, then we further require f0(x) 6= f11 (w0).) We can now construct R2 (similar to Subsubcase 2.2.1) by taking a fault-free Hamiltonian path from f0(x) to f1j1(w 0) in CG1,j1 ([jr=1Fr) {f0(u), f1(f0(u)), . . . , f 1 j1(y)}, followed by P tj . Now consider G0 F0. Recall that |F0|  1. If there is a z such that (u, z) is an arc in G0 and {f1n1(z), (f 1 n1(z), z)} \ F 6= ;, then set F 00 = F0 [ {(u, z)}; otherwise, F 00 = F0. Now we find a Hamiltonian path P0 from s to x in G0 F 00. Let (u, v) on P0. By construction, f1n1(v), (f 1 n1(v), v) 62 F . We can now construct R1, R3, Q0, Q 0 0 as in Subsubcase 2.2.1 with the following extra condition for choosing v when j = n 2: fn2(w) 6= f1n1(v). We also note that R3 starts at fj(w) rather than fj(y). (See Figure 6.) Subsubcase 2.2.4: |Fq| = k 1 for some q = 1, 2, . . . , j 1. One can adapt the construction in Subsubcase 2.2.3 to cover this case. For completeness, we describe the procedure. We note that there is at most one fault not in Fq . We pick two (distinct) vertices a and b in GqFq such that (f10 (· · · f 1 q1(a)), . . . , f 1 j1(a), a) is fault-free and (b, fq(b)) is fault-free. If q = j 1, we further require that fq(b) 6= t. Let Pq be a Hamiltonian path from a to b in Gq Fq . We find an arc (w,w0) on Pq such that (w, fq(w)) and (f1q1(w 0), w0) are fault-free. (We allow a = w or w0 = b.) If j = 1, we further require f 1 0 (w 0) 6= s. Let P yq and P tq be the subpaths of Pj from a to w and from w0 to b, respectively. Let u = f10 (· · · f 1 q1(a)) and R1 be (u, . . . , f 1 q1(a), a), followed by P aq and (w, fj(w)). If j = 1, then let x = f10 (w0). Otherwise, we pick x in G0 (F0 [ {s}) such that (x, f0(x)) is fault-free. We can now construct R2 by taking a fault-free Hamiltonian path from f0(x) to f1j1(w 0) in CG1,q1 ([jr=1Fr) {f0(u), f1(f0(u))), . . . , f 1 j1(a)}, followed by P tq . The rest is the same as Subsubcase 2.2.3. We remark that the main reason that the argument given in Subsubcase 2.2.3 is not valid for k = 2 is because two vertices may be removed from a Gi and hence R2 cannot be constructed as Gi is only 1-Hamiltonian connected. The same problem occurs for the other subcases. In fact, we did not notice this gap and gave this proof for k 2 in an earlier draft. Fortunately for us, the anonymous referee noticed the error. We do not see an easy way to repair this gap. We note that for Subsubcase 2.2.3, the path R1 and R2 together span several Gi’s, and in general, R1 only covers one vertex of such Gi’s. One idea is to be less restrictive in using the two paths covering such Gi’s. (For example, the vertices in Gj Fj in Subsubcase 2.2.3 are spanned by two paths R2 and R3, with each path covering possibly more than one vertex of Gi Fi.) Due to the distribution of s and t and the two vertices in F , there are 8 cases to consider with additional “boundary” subcases in each. We feel that a full discussion adds minimal value. So we choose to present the result for k 3 only. C.-N. Hung et al.: Fault-Hamiltonicity of Cartesian products of directed cycles 15 y t s uv R2 R1 Q0 Q0 f -1 (v)n-1 f (w)j f (x)0 f (u)0 ‘ x R3 f -1(j-1)(w’) w’ w Figure 6: The Hamiltonian path of Subsubcase 2.2.3. 16 Art Discrete Appl. Math. 1 (2018) #P1.06 3 Analyzing the conditions in Theorem 2.2 In Theorem 2.2, one of the conditions is the requirement that N k + 4 for (k + 1)- Hamiltonicity and N k + 5 for k-Hamiltonian connectedness. We are unsure whether this condition can be relaxed. However, we do know that the result for k-Hamiltonian connectedness does not hold if N = k + 2. We choose G = Kk+2, the complete di- rected graph on k + 2 vertices. Clearly it is (k + 1)-regular and one can check that it is k-Hamiltonian and (k 1)-Hamiltonian connected. Then consider H = G⇤C3. Let F be k vertices in G0, s be a vertex in G0 F and t = f(s). Then it is clear that there is no Hamiltonian path from s to t. One may wonder whether there is a counterexample N = k + 3? An obvious choice is to let G be the directed graph obtained from the com- plete graph Kk+3, where k is even, by deleting a perfect matching (and treat the resulting graph as a directed graph). So G is (k + 1)-regular. If G is k-Hamiltonian and (k 1)- Hamiltonian connected, then we have a counterexample. Unfortunately, this is not true as if k = 2i 1, then by deleting appropriate k 1 vertices from G, we have a 4-cycle which is not Hamiltonian connected. We now consider the condition on k. As we pointed out earlier that k = 0 is not applicable, that is, G needs to be at least 2-regular. We have the condition k 2 (that is, G is at least 3-regular) in the statement. In the proof, we did use this assumption; for example, we used it in Case 2 in proving that H is (k + 1)-Hamiltonian. This is not to say that the result is not true for k = 1. On the other hand, we know of no 2-regular, 1-Hamiltonian and Hamiltonian-connected directed graphs. We have already commented on the condition of k 3 for the k-Hamiltonian connectedness portion of the theorem. Finally, there is the condition on n. In an undirected graph, a cycle must have at least three vertices. By the same convention, one usually requires a directed cycle in a directed graph to have at least three vertices; thus the condition n 3. However, some authors do consider the two arcs (x, y) and (y, x) to form a directed cycle of length two. In any case, one may consider two directed graphs G and H with the same number of vertices and construct a new directed graph by two set of matchingss that match the vertices of G with the vertices of H and orient the edges in the first set from G to H and vice versa for the second set. One can ask if both G and H have “strong” Hamiltonian properties, does the resulting graph have “strong” Hamiltonian properties. One can apply similar analysis as in the proof of Theorem 2.2 for this problem. We further remark that Theorem 2.2 seeks the strongest possible property, that is, for a (k+1)-regular graph G to be k-Hamiltonian and (k1)-Hamiltonian connected, and then consider an n-G-directed graph. The proof of Theorem 2.2 mainly relies on G being k- Hamiltonian and (k 1)-Hamiltonian connected, and not G being k-regular. So our proof is applicable to establish the following: Let k 2 and n 3. Let 1  r  k. Suppose the class of directed graphs G is (k+1)-regular, r-Hamiltonian, (r1)-Hamiltonian connected and of order N . Let H be an n-G-directed graph. Then H is (k + 2)-regular, (r + 1)- Hamiltonian if N k + 4 and r-Hamiltonian connected if N k + 5. References [1] C.-W. Cheng, C.-W. Lee and S.-Y. Hsieh, Conditional edge-fault Hamiltonicity of Cartesian product graphs, IEEE Trans. Parallel Distrib. Syst. 24 (2013), 1951–1960, doi:10.1109/tpds. 2012.304. C.-N. Hung et al.: Fault-Hamiltonicity of Cartesian products of directed cycles 17 [2] S. J. Curran and D. Witte, Hamilton paths in Cartesian products of directed cycles, in: B. R. Alspach and C. D. Godsil (eds.), Cycles in Graphs, North-Holland, Amsterdam, volume 115 of North-Holland Mathematics Studies, pp. 35–74, 1985, doi:10.1016/s0304-0208(08)72996-7. [3] R.-X. Hao, R. Zhang, Y.-Q. Feng and J.-X. Zhou, Hamiltonian cycle embedding for fault tol- erance in balanced hypercubes, Appl. Math. Comput. 244 (2014), 447–456, doi:10.1016/j.amc. 2014.07.015. [4] T.-Y. Ho, C.-K. Lin, J. J. M. Tan and L.-H. Hsu, Fault-tolerant Hamiltonian connectivity of the WK-recursive networks, Inf. Sci. 271 (2014), 236–245, doi:10.1016/j.ins.2014.02.087. [5] S.-Y. Hsieh, G.-H. Chen and C.-W. Ho, Fault-free Hamiltonian cycles in faulty arrangement graphs, IEEE Trans. Parallel Distrib. Syst. 10 (1999), 223–237, doi:10.1109/71.755822. [6] S.-Y. Hsieh and C.-W. Lee, Conditional edge-fault Hamiltonicity of matching composition net- works, IEEE Trans. Parallel Distrib. Syst. 20 (2009), 581–592, doi:10.1109/tpds.2008.123. [7] H.-C. Hsu, T.-K. Li, J. J. M. Tan and L.-H. Hsu, Fault Hamiltonicity and fault Hamiltonian connectivity of the arrangement graphs, IEEE Trans. Computers 53 (2004), 39–53, doi:10. 1109/tc.2004.1255789. [8] L.-H. Hsu and C.-K. Lin, Graph Theory and Interconnection Networks, CRC Press, Boca Ra- ton, Florida, 2009. [9] L. E. Penn and D. Witte, When the Cartesian product of two directed cycles is hypo- Hamiltonian, J. Graph Theory 7 (1983), 441–443, doi:10.1002/jgt.3190070409. [10] I. A. Stewart, Sufficient conditions for Hamiltonicity in multiswapped networks, J. Parallel Distrib. Comput. 101 (2017), 17–26, doi:10.1016/j.jpdc.2016.10.015. [11] T.-Y. Sung, C.-Y. Lin, Y.-C. Chuang and L.-H. Hsu, Fault tolerant token ring embedding in double loop networks, Inf. Process. Lett. 66 (1998), 201–207, doi:10.1016/s0020-0190(98) 00066-0. [12] W. T. Trotter, Jr. and P. Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978), 137–142, doi:10.1002/jgt.3190020206. [13] S. Viswanathan, É. Czabarka and A. Sengupta, On fault-tolerant embedding of Hamiltonian circuits in line digraph interconnection networks, Inf. Process. Lett. 57 (1996), 265–271, doi: 10.1016/0020-0190(96)00011-7. [14] Y. Yang and S. Wang, Fault-free Hamiltonian cycles passing through a linear forest in ternary n-cubes with faulty edges, Theoret. Comput. Sci. 491 (2013), 78–82, doi:10.1016/j.tcs.2012. 10.048. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.07 https://doi.org/10.26493/2590-9770.1244.720 (Also available at http://adam-journal.eu) Hamiltonicity of token graphs of fan graphs ⇤ Luis Manuel Rivera Unidad Académica de Matemáticas, Universidad Autónoma de Zacatecas, Calzada Solidaridad entronque Paseo a la Bufa, Zacatecas, Zac. CP. 98000, México Ana Laura Trujillo-Negrete Departamento de Matemáticas, Cinvestav, Av. IPN #2508, Col. San Pedro Zacatenco, México, Cd. de México, CP. 07360, México Received 25 July 2017, accepted 8 October 2017, published online 13 February 2018 Abstract In this note we show that the token graphs of fan graphs are Hamiltonian. This result provides another proof of the Hamiltonicity of Johnson graphs and also extends previous results obtained by Mirajkar and Priyanka on the token graphs of wheel graphs. Keywords: Token graphs, Johnson graphs, Hamiltonian graphs. Math. Subj. Class.: 05C38, 05C45 1 Introduction Let G be a simple graph of order n and let k be an integer such that 1  k  n 1. The k-token graph, or symmetric kth power, of G is the graph G(k) whose vertices are the k-subsets of V (G) and two vertices are adjacent in G(k) if their symmetric difference is an edge of G. A classical example is the Johnson graph J(n, k), which is isomorphic to the k-token graph of the complete graph Kn. This class of graphs is widely studied and has connections with coding theory [7, 9, 11, 13, 15] (another connection of token graphs with coding theory was showed in [18]). The definition of k-token graphs (without a name) appeared in a work of Rudolph [17], in connection with problems in quantum mechanics and with the graph isomorphism prob- lem. Rudolph presented examples of cospectral graphs G and H such that their correspond- ing 2-token graphs are not cospectral. Audenaert et al. [3], proved that the 2-token graphs ⇤Part of this work was made when the second author was a master student at Universidad Autónoma de Zacatecas. The authors would like to thank the anonymous reviewer for his/her corrections and suggestions. E-mail addresses: luismanuel.rivera@gmail.com (Luis Manuel Rivera), ltrujillo@math.cinvestav.mx (Ana Laura Trujillo-Negrete) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.07 of strongly regular graphs with the same parameters are cospectral, and suggested that for a given positive integer k there exists infinitely many pairs of non-isomorphic graphs with cospectral k-token graphs. This conjecture was proved by Barghi and Ponomarenko [16] and, independently, by Alzaga et al. [2]. Later, Fabila-Monroy et al. [8] reintroduced the k-token graphs as part of several models of swapping in the literature [19, 20], and studied some properties of these graphs: connectivity, diameter, cliques, chromatic number, Hamil- tonian paths and Cartesian product of token graphs. This line of research was continued by Carballosa et al. [4] who studied regularity and planarity, de Alba et al. [6], who presented some results about independence and matching numbers, and Mirajkar et al. [14], who stud- ied some covering properties of token graphs. Finally, Leaños and Trujillo-Negrete [12] proved a conjecture of Fabila-Monroy et al. [8] about the connectivity of token graphs and de Alba et al. [5] classified the triangular graphs (in other words, the 2-token graphs of complete graphs) that are Cohen-Macaulay. A graph is Hamiltonian if it contains a Hamiltonian cycle. It is well known that J(n, k) is Hamiltonian [10, 21], in fact, it is Hamiltonian connected [1]. As was noted in [8], the existence of a Hamiltonian cycle in G does not imply that G(k) contains a Hamiltonian cycle. For example, if k is even then K(k)m,m is not Hamiltonian. We are interested in graphs G such that its token graphs are Hamiltonian. The fan graph Fn is the join of graphs K1 and Pn1. In this note we show that the token graphs of fan graphs are Hamiltonian. Our result provides another proof that J(n, k) is Hamiltonian, and also extends some of the results obtained by Mirajkar and Priyanka Y. B [14] about the Hamiltonicity of the token graphs of wheel graphs. 2 Main result First we present some definitions and notations. For vertices u, v in graph G we write u ⇠ v to mean that u and v are adjacent vertices in G. We write G ' G0 if G and G0 are isomorphic graphs. A spanning subgraph of G is a subgraph H such that V (H) = V (G). The following proposition is obvious. Proposition 2.1. If H is a spanning subgraph of G and H is Hamiltonian then G is Hamil- tonian. One of the main properties of token graphs is that G(1) and G are isomorphic. More- over, G(k) ' G(nk) for any k 2 {1, . . . , n1}. Another known property of token graphs is the following. Proposition 2.2. If H is a subgraph of G then H (k) is a subgraph of G (k) . Even more, if H is a spanning subgraph of G then H (k) is a spanning subgraph of G (k) . For a fan graph Fn we assume that the vertices of Pn1 are {1, . . . , n 1} and the vertex in K1 is labeled as n. For vertex A = {a1, . . . , ak} of F (k)n we use the convention that a1 < · · · < ak. The main result of this note is the following. Theorem 2.3. Let n and k be positive integers with n 3 and 1  k  n 1. Then F {k}n is Hamiltonian. Proof. For k = 1, F (1)n ' Fn which is Hamiltonian so in the rest of the proof we as- sume that k 2. We will show that F (k)n has a Hamiltonian cycle such that the vertices L. M. Rivera and A. L. Trujillo-Negrete: Hamiltonicity of token graphs of fan graphs 3 {n k, n k + 1, . . . , n 2, n} and {n k, n k + 1, . . . , n 2, n 1} are adjacent in the cycle. The sequence of vertices {1, 3}{1, 2}{2, 3}{1, 3} is a Hamiltonian cycle in F (2) 3 . The proof for n 4 is by induction on k. First we show the case k = 2 and n 4. The sequence of vertices {1, n 1}{1, n}{1, n 2}{1, n 3} . . . {1, 3}{1, 2} {2, n}{2, n 1}{2, n 2}{2, n 3} . . . {2, 4}{2, 3} ... {n 3, n}{n 3, n 1}{n 3, n 2} {n 2, n}{n 2, n 1} {n 1, n} {1, n 1} is a Hamiltonian cycle in F (2)n , where vertices {n2, n1} and {n2, n} are adjacent in the cycle. We assume as induction hypothesis that F (k 0) n0 satisfies the conditions whenever k 0 < k and n0 > k0. Claim. Let Si be the subgraph of F (k) n induced by the vertex set Vi = n {a1, . . . , ak} 2 V (F (k)n ) : a1 = i o , with 1  i  n k. Then Si ' F (k1)ni . Proof of Claim. Suppose that V (Fni) = {i + 1, . . . , n} with V (Pni1) = {i + 1, . . . , n 1} and n the vertex of K1. Then the function A 7! A \ {i} is a graph isomorphism between Si and F (k1) ni . We identify Si with F (k1) ni using the isomorphism given in the proof of the claim. By induction there exists a Hamiltonian cycle Ci in Si, where vertices Xi := {i, n k + 1, . . . , n 2, n 1} and Yi := {i, n k + 1, . . . , n 2, n} are adjacent in Ci, for 1  i  n k. Let Pi be the Hamiltonian subpath of Ci from Xi to Yi, for 1  i  n k. Let Z denote the vertex {n k + 1, n k + 2, . . . , n 1, n}. Therefore Vnk+1 = {Z}. Let Di = {n k, n k + 1, . . . , n 1, n} \ {i}, with n k + 1  i  n. Then the vertex set Vnk of Snk is {Dn, Dn1, . . . , Dnk+1}. Also, we have Xnk = Dn and Ynk = Dn1. Let Q = Dn2Dn3 . . . Dnk+2Dnk+1, which, in fact, is a path in Snk because Di4Di1 = {i1, i}, for nk+2  i  n2. Now, Xnk4Dn2 = {n 2, n} Ynk4Dn2 = {n 2, n 1} Z4Dnk+1 = {n k, n k + 1} 4 Art Discrete Appl. Math. 1 (2018) #P1.07 and hence Xnk ⇠ Dn2, Ynk ⇠ Dn2, Dnk+1 ⇠ Z, in F (k)n . Notice that Xi4Xi+1 = {i, i+1} and Yi4Yi+1 = {i, i+1}, for 1  i  nk1, and X14Z = {1, n}. Therefore we can define a Hamiltonian cycle C in F (k)n as X1 ! P1 Y1Y2 ! P2 X2 . . . X(nk1) ! Pnk1 Y(nk1)Y(nk)X(nk)Dn2 ! Q Dnk+1ZX1, if n k is even, and X1 ! P1 Y1Y2 ! P2 X2 . . . Y(nk1) ! Pnk1 X(nk1)X(nk)Y(nk)Dn2 ! Q Dnk+1ZX1, if n k is odd. Furthermore {n k, n k+ 1, ..., n 2, n 1} = Xnk ⇠ Ynk = {n k, n k+ 1, ..., n 2, n}, in C, as desired. The wheel graph Wn is the joint graph of K1 and Cn1. It is known that Johnson graphs [10, 21] and the k-token graphs of wheel graphs [14] are Hamiltonian, the following corollary provides another proof of this facts. Corollary 2.4. If Fn is a spanning subgraph of G then G (k) is Hamiltonian. In particular the Johnson graphs and the k-token graphs of wheel graphs are Hamiltonian. Proof. As Fn is a spanning subgraph of G then F (k) n is a spanning subgraph of G(k) by Proposition 2.2. The k-token graph of Fn is Hamiltonian by Theorem 2.3 and hence G(k) is Hamiltonian by Proposition 2.1. In particular Fn is a spanning subgraph of Wn and Kn. References [1] B. Alspach, Johnson graphs are Hamilton-connected, Ars Math. Contemp. 6 (2013), 21–23, http://amc-journal.eu/index.php/amc/article/view/291. [2] A. Alzaga, R. Iglesias and R. Pignol, Spectra of symmetric powers of graphs and the Weisfeiler- Lehman refinements, J. Comb. Theory Ser. B 100 (2010), 671–682, doi:10.1016/j.jctb.2010.07. 001. [3] K. Audenaert, C. Godsil, G. Royle and T. Rudolph, Symmetric squares of graphs, J. Comb. Theory Ser. B 97 (2007), 74–90, doi:10.1016/j.jctb.2006.04.002. [4] W. Carballosa, R. Fabila-Monroy, J. Leaños and L. M. Rivera, Regularity and planarity of token graphs, Discuss. Math. Graph Theory 37 (2017), 573–586, doi:10.7151/dmgt.1959. [5] H. de Alba, W. Carballosa, D. Duarte and L. M. Rivera, Cohen-Macaulayness of triangular graphs, Bull. Math. Soc. Sci. Math. Roumanie 60 (2017), 103–112, http://ssmr.ro/ bulletin/volumes/60-2/node2.html. L. M. Rivera and A. L. Trujillo-Negrete: Hamiltonicity of token graphs of fan graphs 5 [6] H. de Alba, W. Carballosa, J. Leaños and L. M. Rivera, Independence and matching numbers of some token graphs, 2016, arXiv:1606.06370 [math.CO]. [7] T. Etzion and S. Bitan, On the chromatic number, colorings, and codes of the Johnson graph, Discrete Appl. Math. 70 (1996), 163–175, doi:10.1016/0166-218x(96)00104-7. [8] R. Fabila-Monroy, D. Flores-Peñaloza, C. Huemer, F. Hurtado, J. Urrutia and D. R. Wood, Token graphs, Graphs Combin. 28 (2012), 365–380, doi:10.1007/s00373-011-1055-9. [9] R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inf. Theory 26 (1980), 37–43, doi:10.1109/tit.1980.1056141. [10] H. R. Ho, Hamiltonicity of the graph G(n, k) of the Johnson scheme, Jur- nal Informatika 3 (2007), 41–47, http://majour.maranatha.edu/index.php/ jurnal-informatika/article/view/263. [11] S. M. Johnson, A new upper bound for error-correcting codes, IRE Trans. Inf. Theory 8 (1962), 203–207, doi:10.1109/tit.1962.1057714. [12] J. Leaños and A. L. Trujillo-Negrete, The connectivity of token graphs, submitted. [13] R. A. Liebler and C. E. Praeger, Neighbour-transitive codes in Johnson graphs, Des. Codes Cryptogr. 73 (2014), 1–25, doi:10.1007/s10623-014-9982-0. [14] K. G. Mirajkar and P. Y. B, Traversability and covering invariants of token graphs, International J. Math. Combin. 10 (2016), 132–138, http://fs.gallup.unm.edu/IJMC-2-2016. pdf. [15] M. Neunhöffer and C. E. Praeger, Sporadic neighbour-transitive codes in Johnson graphs, Des. Codes Cryptogr. 72 (2014), 141–152, doi:10.1007/s10623-013-9853-0. [16] A. Rahnamai Barghi and I. Ponomarenko, Non-isomorphic graphs with cospectral symmet- ric powers, Electron. J. Comb. 16 (2009), #R120, http://www.combinatorics.org/ Volume_16/Abstracts/v16i1r120.html. [17] T. Rudolph, Constructing physically intuitive graph invariants, 2002, arXiv:quant-ph/0206068. [18] N. J. A. Sloane, Sequence A085680 in The On-Line Encyclopedia of Integer Sequences, pub- lished electronically at https://oeis.org. [19] J. van den Heuvel, The complexity of change, in: S. R. Blackburn, S. Gerke and M. Wildon (eds.), Surveys in Combinatorics 2013, Cambridge University Press, Cambridge, volume 409 of London Mathematical Society Lecture Note Series, pp. 127–160, 2013, doi:10.1017/ cbo9781139506748.005. [20] K. Yamanaka, E. D. Demaine, T. Ito, J. Kawahara, M. Kiyomi, Y. Okamoto, T. Saitoh, A. Suzuki, K. Uchizawa and T. Uno, Swapping labeled tokens on graphs, in: A. Ferro, F. Luccio and P. Widmayer (eds.), Fun with Algorithms, Springer, Cham, Switzerland, volume 8496 of Lecture Notes in Computer Science, pp. 364–375, 2014, doi:10.1007/978-3-319-07890-8 31. [21] F. Zhang, G. Lin and R. Cheng, Some distance properties of the graph G(n, k) of John- son scheme, 1997, talk presented at the International Congress in Algebras and Com- binatorics (Hong Kong, 19–23 August 1997), https://www.math.cuhk.edu.hk/ conference/icac1997/. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.08 https://doi.org/10.26493/2590-9770.1245.98c (Also available at http://adam-journal.eu) A short note on undirected Fitch graphs⇤ Marc Hellmuth † Institute of Mathematics and Computer Science, University of Greifswald, Walther-Rathenau-Straße 47, D-17487 Greifswald, Germany Yangjing Long School of Mathematics and Statistics, Central China Normal University, No. 152, Luoyu Road, Wuhan, Hubei, P. R. China Manuela Geiß , Peter F. Stadler ‡ Bioinformatics Group, Department of Computer Science, Universität Leipzig, Härtelstrasse 16-18, D-04107 Leipzig, Germany Received 5 December 2017, accepted 16 February 2018, published online 7 March 2018 Abstract Fitch graphs have been introduced as a model of xenology relationships in phyloge- nomics. Directed Fitch graphs G = (X,E) are di-graphs that are explained by {0, 1}- edge-labeled rooted trees with leaf set X: there is an arc xy 2 E if and only if the unique path in T that connects the least common ancestor lca(x, y) of x and y with y contains at least one edge with label 1. In this contribution, we consider the undirected version of Fitch’s xenology relation, in which x and y are xenologs if and only if the unique path be- tween x and y in T contains an edge with label 1. We show that symmetric Fitch relations coincide with class of complete multipartite graph and thus cannot convey any non-trivial phylogenetic information. Keywords: Labeled trees, forbidden subgraphs, phylogenetics, xenology, Fitch graph. Math. Subj. Class.: 05C75, 05C05, 92B10 ⇤This work is supported in part by the BMBF-funded project “Center for RNA-Bioinformatics” (031A538A, de.NBI/RBC) and the German Academic Exchange Service (PROALMEX, grant no. 57274200). †MH is also affiliated with the Center for Bioinformatics, Saarland University, Building E 2.1, P.O. Box 151150, D-66041 Saarbrücken, Germany. ‡PFS is also affiliated with the Interdisciplinary Center for Bioinformatics, the German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, the Competence Center for Scalable Data Services and Solutions Dresden-Leipzig, the Leipzig Research Center for Civilization Diseases, and the Centre for Biotechnology and Biomedicine at Leipzig University; the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany; the Institute for Theoretical Chemistry, University of Vienna, Vienna, Austria; the Center of noncoding RNA in Health and Technology (RTH) at the University of Copenhagen; and the Santa Fe Institute, Santa Fe, NM. cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.08 Fitch graphs [4] form a class of directed graphs that is derived from rooted, {0, 1}-edge- labeled trees T in the following manner: The vertices of the Fitch graph are the leaves of T . Two distinct leaves x and y of T are connected by an arc (x, y) from x to y if and only if there is at least one edge with label 1 on the (unique) path in T that connects the least common ancestor lca(x, y) of x and y with y. Fitch graphs model “xenology”, an important binary relation among genes, was introduced by Walter M. Fitch [2]. Interpreting T as a phylogenetic tree and 1-edges as horizontal gene transfer events, the arc (x, y) in the Fitch graph encodes the fact that y is xenologous with respect to x. A complete characterization of directed Fitch graphs is given in [4] in terms of the eight forbidden induced subgraphs shown in Figure 1. F1 F2 F3 F4 F5 F6 F8F7 Figure 1: Shown are the eight forbidden induced subgraphs F1, . . . , F8 of Fitch graphs. Theorem 0.1 ([4]). A digraph G = (X,E) is a directed Fitch graph if and only if it does not contain one the graphs F1, . . . , F8 in Figure 1 as an induced subgraph. It can be decided in O(|X| + |E|) time whether G is a directed Fitch graph. In the positive case, there is a unique least-resolved tree (T,) explaining G, which also can be constructed in linear time. It is natural to consider also the symmetrized version of this relationship, i.e., to inter- pret {x, y} as a xenologous pair whenever the evolutionary history separated x and y by at least one horizontal transfer event. In mathematical terms, this idea is captured by: Definition 0.2. Let T be a rooted tree with leaf set X and let : E(T ) ! {0, 1}. Then the undirected Fitch graph F explained by (T,) has vertex set X and edges {x, y} 2 E(F ) if and only if the (unique) path from x to y in T contains at least one edge e with (e) = 1. A graph F is an undirected Fitch graph if and only if it is explained in this manner by some edge-labeled rooted tree (T,). Undirected Fitch graphs are closely related to their directed counterparts. Since the path } connecting two leaves x and y is unique and contains their least common ancestor lca(x, y), there is a 1-edge along } if and only if there is a 1-edge on the path between x and lca(x, y) or between lca(x, y) and y. The undirected Fitch graph is therefore the underlying undirected graph of the directed Fitch graph, i.e., it is obtained from the directed version by ignoring the direction of the arcs. The undirected Fitch graphs form a heritable family, i.e., if F is an undirected Fitch graph, so are all its induced subgraphs. This is an immediate consequence of the fact that E-mail addresses: mhellmuth@mailbox.org (Marc Hellmuth), yjlong@sjtu.edu.cn (Yangjing Long), manuela@bioinf.uni-leipzig.de (Manuela Geiß), studla@bioinf.uni-leipzig.de (Peter F. Stadler) M. Geiß, M. Hellmuth, Y. Long and P. F. Stadler: Undirected Fitch graphs 3 directed Fitch graphs are a heritable family of digraphs [4]. The fact can also be obtained directly by considering the restriction of T to a subset of leaves. This obviously does not affect the paths or their labeling between the remaining vertices. Clearly F does not depend on which of the non-leaf vertices in T is the root. Fur- thermore, a vertex v with only two neighbors and its two incident edges e0 and e00 can be replaced by a single edge e. The new edge is labeled (e) = 0 if both (e0) = (e00) = 0, and (e) = 1 otherwise. These operations do not affect the undirected Fitch graph. Hence, we can replace the rooted tree T by an unrooted tree in Definition 0.2 and assume that all non-leaf edges have at least degree 3. To avoid trivial cases we assume throughout that T has at least two leaves and hence a Fitch graph has at least two vertices. Lemma 0.3. If G is an undirected Fitch graph, then F does not contain K1 ￿K2 as an induced subgraph. In particular every undirected Fitch graph is a complete multipartite graph. Proof. There is a single unrooted tree with three leaves, namely the star S3, which ad- mits four non-isomorphic {0, 1}-edge labelings defined by the number N of 1-edges. The undirected Fitch graphs FN are easily obtained. In the absence of 1-edges, F0 = K3 is edge-less. For N = 2 and N = 3 there is a 1-edge along the path between any two leaves, i.e., F2 = F3 = K3. For N = 1 one leaf is connected to the other two by a path in S3 with an 1-edge; the path between the latter two leaves consists of two 0-edges, hence F1 = P3, the path of length two. Hence, only three of the four possible undirected graphs on three vertices can be realized, while K1 ￿K2 is not an undirected Fitch graph. By heredity, K1 ￿K2 is therefore a forbidden induced subgraph for the class of undirected Fitch graphs. Finally, it is well-known that the class of graphs that do not contain K1 ￿K2 as an induced subgraph are exactly the complete multipartite graphs, see e.g. [8]. We note in passing that the first part of Lemma 0.3 can also be obtained from the eight forbidden graphs on three vertices, using the fact that an undirected Fitch graph is the underlying (undirected) graph of a directed Fitch graph. In order to show that forbidding K1 ￿K2 is also sufficient, we explicitly construct the edge-labeled trees necessary to explain complete multipartite graphs. We start by recall- ing that each complete multipartite graph Kn1,...,nk is determined by its independent sets V1, . . . , Vk with |Vi| = ni for 1  i  k. By definition, {x, y} 2 E(Kn1,...,nk) if and only if x 2 Vi and y 2 Vj with i 6= j. In particular, therefore, Kn1,...,nk with at least two vertices is connected if and only if k 2. The complete 1-partite graphs are the edge-less graphs Kn. Since K1 ￿K2 is an induced subgraph of the path on four vertices P4, any graph G that does not contain K1 ￿K2 as an induced subgraph must be P4-free, i.e., a cograph [1]. Cographs are associated with vertex-labeled trees known as cotrees, which in turn are a special case of modular decomposition trees [3]. The cotrees of connected multipartite graphs have a particularly simple shape, illustrated without the vertex labels in Figure 2. The cotree has a root labeled “1” and all inner vertices labeled “0”. Here we do not need the connection between cographs and their cotrees, however. Therefore, we introduce these trees together with an edge-labeling that is useful for our purposes in the following: Definition 0.4. For k = 1, T [n] is the star graph Sn with n leaves. For k 2, the tree T [n1, . . . , nk] has a root r with k children ci, 1  i  k. The vertex ci is a leaf if |Vi| = ni = 1 and has exactly ni children that are leaves if |Vi| = ni 2. For k = 1 all 4 Art Discrete Appl. Math. 1 (2018) #P1.08 edges e of T [n] are labeled ⇤(e) = 0. For k 2 we set ⇤({r, ci}) = 1 for 1  i  k and ⇤(e) = 0 for all edges not incident to the root. 1 2 3 1 2 3 4 5 6 7 4 5 7 6 1 1 1 1 0 0 0 0 0 Figure 2: The complete multipartite graph K3,2,1,1 is the Fitch graph explained by the tree T [3, 2, 1, 1] with edge labeling ⇤ shown with bold numbers 0 and 1. Now we can prove our main result: Theorem 0.5. A graph G is an undirected Fitch graph if and only if it is a complete multipartite graph. In particular, Kn1,...,nk is explained by (T [n1, . . . , nk], ⇤). Proof. Lemma 0.3 implies that an undirected Fitch graph is a complete multipartite graph. To show the converse, we fix an arbitrary complete multipartite graph G = Kn1,...,nk and find an edge-labeled rooted tree (T,⇤) that explains G. For k = 1 it is trivial that (T [n],⇤) explains Kn. For k 2 consider the tree T [n1, . . . nk] with edge labeling ⇤ and let F be the cor- responding Fitch graph. The leaf set of T [n1, . . . nk] is partitioned into exactly k subsets L1, . . . , Lk defined by (a) singletons adjacent to the root and (b) subsets comprising at least two leaves adjacent to the same child ci of the root. Furthermore, we can order the leaf sets so that |Li| = ni. By construction, all vertices within a leaf set Li are connected by a path that does not run through the root and thus, contains only 0-edges, if |Li| > 1 and no edge, otherwise. The Li are independent sets in F . On the other hand any two leaves x 2 Li and y 2 Lj with i 6= j are connected only by path through the root, which contains two 1-edges. Thus x and y are connected by an edge in F . Hence F is a complete multipartite graph of the form K|L1|,...,|Lk| = Kn1,...,nk . Since Kn1,...,nk is explained by (T [n1, . . . , nk],⇤) for all ni 1 and all k 2, and Kn is explained by (T [n],⇤), we conclude that every complete multipartite graph is a Fitch graph. The converse of Lemma 0.3 does not follow in a straightforward manner from the characterization of directed Fitch graphs in [4]. It is possible to make use of the connection between Fitch graphs and di-cographs [5, 6] to obtain the trees of Definition 0.4. This line of reasoning, however, is neither shorter nor simpler than the direct, elementary proof given above. Complete multipartite graphs G = (V,E) obviously can be recognized in O(|V |2) time (e.g., by checking that its complement is a disjoint union of complete graphs), and even in O(|V |+ |E|) time by explicitly constructing its modular decomposition tree [7]. Given the tree T [n1, . . . , nk], the canonical edge labeling ⇤ is then assigned in O(|V |) time. A tree (T,) that explains a Fitch graph F is minimum if it has the smallest number of vertices among all trees that explain F . In this case, (T,) is also least-resolved, i.e., the M. Geiß, M. Hellmuth, Y. Long and P. F. Stadler: Undirected Fitch graphs 5 contraction of any edge in (T,) results in a tree that does not explain F . Not surprisingly, the tree T [n1, . . . , nk] is almost minimum in most, and minimum in some cases: Since the vertices of the Fitch graph must correspond to leaves of the tree, T [n1, . . . , nk] is necessar- ily minimum whenever it is a star, i.e., for T [n] and T [1, . . . , 1]. In all other cases, its only potentially “superfluous” part is its root. Indeed, exactly one of the edges connecting the root with a non-leaf neighbor can be contracted without changing the corresponding Fitch graph. It is clear that this graph is minimal: The leaf sets Li must be leaves of an induced subtree without an intervening 1-edge. Having all vertices of Li adjacent to the same vertex is obviously the minimal choice. Since the Li must be separated from all other leaves by a 1-edge, at least one incident edge of ci must be a 1-edge. Removing all leaves incident to a 0-edge results in a tree with at least k vertices that must contain at least k1 1-edges, since every path between leaves in this tree must contain a 1-edge. The contraction of exactly one of the k 1-edges incident to the root r in T [n1, . . . , nk] indeed already yields a minimal tree. In general, the minimal trees are not unique, see Figure 3. 1 2 3 45 1 5 32 1 1 0 0 0 (T, *) 0 4 1 432 00 1 11 (T[2,2,1], *) 5 00 1 32 00 1 1 (T*[2,2,1], *) 4 5 00 32 00 1 (T', *) 1 54 1 00 Figure 3: The non-isomorphic trees (T,⇤), (T 0,⇤) (T [2, 2, 1],⇤), and (T ⇤[2, 2, 1],⇤) all explain the same complete multipartite graph K2,2,1. Three of these trees have the smallest possible number (7) of vertices and thus are minimal. These can be obtained from the tree (T [2, 2, 1],⇤) specified in Definition 0.2 by contraction of one of its inner 1-edges and possibly re-rooting the resulting tree. It may be worth noting that Kn1,...,nk can also be explained by binary trees. To see this, we convert a tree (T [n1, . . . , nk],⇤) into a binary tree in two simple steps. First, each group of ni > 1 leaves with a common parent are replaced by an arbitrary binary tree with the same leaf set and all edges labeled 0. Second, the star consisting of the root and all its children C is replaced by an arbitrary rooted binary tree with leaf set C and all edges labeled 1. It is obvious that neither of the operations affects the graph that is explained. The practical implication of Theorem 0.5 in the context of phylogenetic combinatorics is that the mutual xenology relation cannot convey any interesting phylogenetic informa- tion: Since the undirected Fitch graphs are exactly the complete multipartite graphs, which in turn are completely defined by their independent sets, the only insight we can gain by considering mutual xenology is the identification of the maximal subsets of taxa that have not experienced any horizontal transfer events among them. References [1] D. G. Corneil, H. Lerchs and L. Stewart Burlingham, Complement reducible graphs, Discrete Appl. Math. 3 (1981), 163–174, doi:10.1016/0166-218x(81)90013-5. [2] W. M. Fitch, Homology: a personal view on some of the problems, Trends Genet. 16 (2000), 227–231, doi:10.1016/s0168-9525(00)02005-9. 6 Art Discrete Appl. Math. 1 (2018) #P1.08 [3] T. Gallai, Transitiv orientierbare Graphen, Acta Math. Acad. Sci. Hungar. 18 (25–66), 1967, doi:10.1007/bf02020961. [4] M. Geiß, J. Anders, P. F. Stadler, N. Wieseke and M. Hellmuth, Reconstructing gene trees from Fitch’s xenology relation, J. Math. Biol., to appear. [5] M. Hellmuth, M. Hernandez-Rosales, Y. Long and P. F. Stadler, Inferring phylogenetic trees from the knowledge of rare evolutionary events, J. Math. Biol. (2017), doi:10.1007/ s00285-017-1194-6. [6] M. Hellmuth, P. F. Stadler and N. Wieseke, The mathematics of xenology: di-cographs, symbolic ultrametrics, 2-structures and tree-representable systems of binary relations, J. Math. Biol. 75 (2017), 299–237, doi:10.1007/s00285-016-1084-3. [7] R. M. McConnell and J. P. Spinrad, Modular decomposition and transitive orientation, Discrete Math. 201 (1999), 189–241, doi:10.1016/s0012-365x(98)00319-7. [8] I. E. Zverovich, Near-complete multipartite graphs and forbidden induced subgraphs, Discrete Math. 207 (1999), 257–262, doi:10.1016/s0012-365x(99)00050-3. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.09 https://doi.org/10.26493/2590-9770.1257.dda (Also available at http://adam-journal.eu) On a conjecture about the ratio of Wiener index in iterated line graphs ⇤ Katarı́na Hriňáková , Martin Knor Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05, Bratislava, Slovakia Riste Škrekovski FMF, University of Ljubljana, 1000 Ljubljana and Faculty of Information Studies, 8000 Novo Mesto and FAMNIT, University of Primorska, 6000 Koper, Slovenia Received 9 January 2017, accepted 30 May 2018, published online 23 July 2018 Abstract Let G be a graph. Denote by W (G) its Wiener index and denote by Li(G) its i- iterated line graph. Dobrynin and Mel’nikov proposed to estimate the extremal values for the ratio Rk(G) = W (Lk(G))/W (G) for k 1. Motivated by this we study the ratio for higher k’s. We prove that among all trees on n vertices the path Pn has the smallest value of this ratio for k 3. We conjecture that this holds also for k = 2, and even more, for the class of all connected graphs on n vertices. Moreover, we conjecture that the maximum value of the ratio is obtained for the complete graph. Keywords: Wiener index, line graph, tree, iterated line graph. Math. Subj. Class.: 05C12, 05C05, 05C76 1 Introduction Let G be a graph. We denote its vertex set and edge set by V (G) and E(G), respectively. For any two vertices u, v let d(u, v) be the distance from u to v. The Wiener index of G, W (G), is defined as W (G) = X u 6=v d(u, v), ⇤The first and second author acknowledge partial support by Slovak research grants VEGA 1/0007/14, VEGA 1/0026/16, VEGA 1/0142/17, APVV-0136-12 and APVV-15-0220. The research was partially supported by Slovenian research agency ARRS, program no. P1-0383. E-mail address: hrinakova@math.sk (Katarı́na Hriňáková), knor@math.sk (Martin Knor), skrekovski@gmail.com (Riste Škrekovski) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.09 where the sum is taken over all unordered pairs of vertices of G. Wiener index was intro- duced by Wiener in [17]. Since it is related to several properties of molecules (see [7]), it is widely studied by chemists. The interest of mathematicians was attracted in 1970’s, when it was reintroduced as the transmission and the distance of a graph, see [16] and [5], respectively. For surveys and some up-to-date papers related to the Wiener index of trees and line graphs see [15, 18] and [2, 8, 13], respectively. By definition, if G has a unique vertex, then W (G) = 0. In this case, we say that the graph G is trivial. The line graph of G, L(G), has vertex set identical with the set of edges of G and two vertices of L(G) are adjacent if and only if the corresponding edges are incident in G. Iterated line graphs are defined inductively as follows: L i(G) = ( G if i = 0, L(Li1(G)) if i > 0. Observe that W (Pn) = ((n1) + · · ·+ 1)+((n2) + · · ·+ 1)+ · · ·+1 = n+1 3 . In the case when a tree contains a small number of branching vertices (i.e., vertices of degree at least three), then it is suitable to use the theorem of Doyle and Graver [4] for computing its Wiener index: Theorem 1.1. Let T be a tree on n vertices. Then W (T ) = ✓ n+ 1 3 ◆ X v2V (T ) X 1i W (T ) otherwise. The above result gives an immediate support to Conjecture 1.5: Corollary 1.7. Let k 4. In the class of trees on n vertices, Rk attains the minimum value for Pn. In this paper we extend the above corollary to the case k = 3. Let H be a tree on six vertices, two of which have degree 3 and the other four have degree 1. Recall that two graphs G1 and G2 are homeomorphic if and only if there is a third graph F , such that both G1 and G2 can be obtained from F by means of edge subdivision. In the proof we will use the following result [9, Corollary 1.6]: Theorem 1.8. Let T be a tree which is not homeomorphic to a path, claw K1,3 or H , and let k 3. Then W (Lk(T )) > W (T ). By Theorem 1.8, to solve the case k = 3, it is sufficient to consider the ratios for paths and trees homeomorphic to the claw K1,3 and H . Note that L3(Pn) = Pn3 if n 4, and we have R3(Pn) = n2 3 n+1 3 = (n 2)(n 3)(n 4) (n+ 1)n(n 1) . In Section 2 we prove the following two results: Theorem 1.9. Let T be a tree on n vertices homeomorphic to K1,3. Then R3(T ) > R3(Pn). Theorem 1.10. Let T be a tree on n vertices homeomorphic to H . Then R3(T ) > R3(Pn). These two results together with Theorem 1.8 and Corollary 1.7 give us the following: Corollary 1.11. Let k 3. Then the path Pn attains the minimum value of Rk in the class of trees on n vertices. 4 Art Discrete Appl. Math. 1 (2018) #P1.09 2 Proofs of Theorems 1.9 and 1.10 Proof of Theorem 1.9. Let Ca,b,c be a tree homeomorphic to the claw K1,3, such that the paths connecting the vertices of degree 1 with the vertex of degree 3 have lengths a, b and c, where a b c 1. The tree Ca,b,c has exactly n = a+b+c+1 vertices, see Figure 1 for C4,3,2. Figure 1: The graph C4,3,2. Further, for i 2 {1, 2, 3} let Vi be the set of vertices of V (L(Ca,b,c)) of degree i. This naturally splits the problem into four cases according to the size of V1. Denote = W (L3(Ca,b,c))W (Ca,b,c). (2.1) In [8], the value of for each of these four cases is evaluated. For the sake of simplicity, let W0 = W (Ca,b,c) and W3 = W (L3(Ca,b,c)). Then = W3 W0 and R3(Ca,b,c) = W3 W0 = W0 + W0 = 1 + W0 . By Theorem 1.1 we have W0 = (a+ b+ c+ 2)(a+ b+ c+ 1)(a+ b+ c)/6 abc. (2.2) We prove that when |V (Ca,b,c)| = |V (Pn)|, that is when n = a + b + c + 1, then R3(Ca,b,c) > R3(Pn). This inequality is equivalent to 1 + W0 > (n 2)(n 3)(n 4) (n+ 1)n(n 1) and after multiplying by denominators also to (n+ 1)n(n 1) +W0 (n+ 1)n(n 1) (n 2)(n 3)(n 4) > 0. (2.3) Since 3 |Vi| 0, there are four cases to consider. Case 1: a, b, c 2. That is, |V1| = 3. In [8] we have = (a+b+c)2 5(ab+ac+bc) + (a+b+c) + 21. (2.4) After substituing (2.4) and (2.2) into (2.3), we get that the left-hand side of (2.3) is equal to K. Hriňáková et al.: On a conjecture about the ratio of Wiener index in iterated line graphs 5 the following expression 1.5abc (a b)2 + (a c)2 + (b c)2 + 44a+ 65a2 + 25.5a3 + 7a4 + 2.5a5 + 44b+ 130ab+ 66.5a2b+ 13a3b+ 7.5a4b+ 65b2 + 66.5ab2 + 12a2b2 + 10a3b2 + 25.5b3 + 13ab3 + 10a2b3 + 7b4 + 7.5ab4 + 2.5b5 + 44c+ 130ac+ 66.5a2c+ 13a3c+ 7.5a4c+ 130bc+ 117abc+ 18a2bc+ 3a3bc+ 66.5b2c+ 18ab2c+ 13b3c+ 3ab3c+ 7.5b4c+ 65c2 + 66.5ac2 + 12a2c2 + 10a3c2 + 66.5bc2 + 18abc2 + 12b2c2 + 10b3c2 + 25.5c3 + 13ac3 + 10a2c3 + 13bc3 + 3abc3 + 10b2c3 + 7c4 + 7.5ac4 + 7.5bc4 + 2.5c5. Since a, b, c 2, the expression 1.5abc (a b)2+(a c)2+(b c)2 and all the isolated terms are nonnegative. Moreover some of the terms, such as 44a for example, are strictly positive. Hence, (2.3) is satisfied, which means that R3(Ca,b,c) > R3(Pa+b+c+1). Observe that the above long expression was obtained from the left-hand side of (2.3) by subtracting 1.5abc (a b)2 + (a c)2 + (b c)2 , which is nonnegative, and then by expanding the difference. Since all the parameters a, b, c are nonnegative, all the co- efficients in the expanded expression are positive and at least one of the terms is strictly positive, (2.3) is satisfied. We will use this way of reasoning especially in the proof of Theorem 1.10, where the expanded expressions are extremely long. Case 2: a, b 2, c = 1. That is, |V1| = 2. In [8] we have 2 = (a+b)2 8ab 5(a+b) + 30. (2.5) After substituing (2.5) and (2.2) into (2.3) and expanding the expression, we get that the left-hand side of (2.3) is equal to 96 + 170a+ 97a2 + 32a3 + 11a4 + 2a5 + 170b+ 164ab+ 43a2b+ 11a3b+ 6a4b+ 97b2 + 43ab2 + 8a3b2 + 32b3 + 11ab3 + 8a2b3 + 11b4 + 6ab4 + 2b5. Hence (2.3) is satisfied and so R3(Ca,b,1) > R3(Pa+b+2). Case 3: a 2, b = c = 1. That is, |V1| = 1. In [8] we have = 6a + 6. After substituing this value of and (2.2) into (2.3) and expanding the expression, we get that the left-hand side of (2.3) is equal to 1.5a5 + 12a4 + 26.5a3 + 60a2 + 300a+ 240. Hence (2.3) is satisfied and so R3(Ca,1,1) > R3(Pa+3). Case 4: a = b = c = 1. That is, |V1| = 0. In this case Ca,b,c = K1,3 has 4 vertices and L 3(K1,3) is a cycle of length 3. Since W (L3(P4)) = 0, we have R3(C1,1,1) > 0 = R3(P4), which establishes this small case, and also the proof of the theorem. 6 Art Discrete Appl. Math. 1 (2018) #P1.09 Proof of Theorem 1.10. Denote by Ha,b,c,d,e a tree homeomorphic to H defined as follows: In Ha,b,c,d,e, the two vertices of degree 3 are joined by a path of length e + 1, e 0. Hence, this path has e vertices of degree 2. Further, at one vertex of degree 3 there start two pendant paths of lengths a and b, where a, b 1, and at the other vertex of degree 3 there start another two pendant paths of lengths c and d, where c, d 1. Thus Ha,b,c,d,e has n = a+ b+ c+ d+ e+2 vertices, out of which two have degree 3, four have degree 1 and the remaining vertices have degree 2, see Figure 2 for H3,3,4,2,2. By symmetry, we may assume that a b, c d, and b d. That is, we assume that the shortest pendant path in Ha,b,c,d,e has length d. Figure 2: The graph H3,3,4,2,2. We proceed analogously as in the proof of Theorem 1.9. Denote = W (L3(Ha,b,c,d,e))W (Ha,b,c,d,e). (2.6) For the sake of simplicity, let W0 = W (Ha,b,c,d,e) and W3 = W (L3(Ha,b,c,d,e)). Then = W3 W0 and again R3(Ha,b,c,d,e) = 1 + W0 . By Theorem 1.1 we have W0 = ✓ a+ b+ c+ d+ e+ 3 3 ◆ ab(c+ d+ e+ 1) cd(a+ b+ e+ 1). (2.7) If e = 0, then we have one vertex of degree 4 in L(Ha,b,c,d,e), while if e 1, then the greatest degree of a vertex in L(Ha,b,c,d,e) is 3. Analogously as in [8], by symmetry we distinguish eleven cases. Five cases with at least one of a, b, c, d greater than or equal to 2 have e 1, five cases with at least one of a, b, c, d greater than or equal to 2 have e = 0, and the last case has all a, b, c, d equal to 1. First we consider the cases with > 0. Claim 1. If > 0, then R3(Ha,b,c,d,e) > R3(Pa+b+c+d+e+2). Proof. Observe that |V (Ha,b,c,d,e)| = |V (Pa+b+c+d+e+2)|. If > 0, then R3(Ha,b,c,d,e) = 1 + W0 > 1. However, R3(Pn) is always smaller than 1. By [8], there are 8 cases (out of the 11) for which in [8] it was proved that > 0 (we remark that P is used instead of in [8]). These are the cases: 1. (case 3 in [8]) a, c 2, b = d = 1, e 1; 2. (case 4 in [8]) a, b 2, c = d = 1, e 1; K. Hriňáková et al.: On a conjecture about the ratio of Wiener index in iterated line graphs 7 3. (case 5 in [8]) a 2, b = c = d = 1, e 1; 4. (case 7 in [8]) a, b, c 2, d = 1, e = 0; 5. (case 8 in [8]) a, c 2, b = d = 1, e = 0; 6. (case 9 in [8]) a, b 2, c = d = 1, e = 0; 7. (case 10 in [8]) a 2, b = c = d = 1, e = 0; 8. (case 11 in [8]) a = b = c = d = 1, e 0. By Claim 1 it suffices to consider the remaining three cases. We proceed analogously as in the proof of Theorem 1.9. Hence, we prove that when |V (Ha,b,c,d,e)| = |V (Pn)|, that is when n = a+ b+ c+ d+ e+ 2, then R3(Ha,b,c,d,e) > R3(Pn). This inequality is equivalent to 1 + W0 > (n 2)(n 3)(n 4) (n+ 1)n(n 1) and after multiplying by denominators also to (n+ 1)n(n 1) +W0 (n+ 1)n(n 1) (n 2)(n 3)(n 4) > 0. (2.8) Now we consider the remaining three cases. Case 1: a, b, c, d 2, e 1. In [8] we have 2 = 7(a+b+c+d+e)2 20(ab+ac+ad+bc+bd+cd) 10(ae+be+ce+de) + 5(a+b+c+d) + 65e+ 234. (2.9) Denote D = 11 cd(a b)2(a+ b) + bd(a c)2(a+ c) + ad(b c)2(b+ c) + bc(a d)2(a+ d) + ac(b d)2(b+ d) + ab(c d)2(c+ d) . Observe that D 0. Now substitute (2.9) and (2.7) into the left-hand side of (2.8) and delete D. When we expand the resulting expression, all the coefficients will be posi- tive. Since the constant term is 708, which is strictly positive, (2.8) is satisfied and so R3(Ha,b,c,d,e) > R3(Pa+b+c+d+e+2). Case 2: a, b, c 2, d = 1, e 1. From [8] we have = 3(a2+b2+c2+e2) 3(ab+ac+bc) + (ae+be) + 2ce 2(a+ b) c+ 28e+ 97. In [8] it was shown that if e 2 then > 0. By Claim 1, R3(Ha,b,c,1,e) > R3(Pa+b+c+e+3) in this subcase, so it suffices to restrict ourselves to e = 1. For e = 1 we obtain = 3(a2+b2+c2) 3(ab+ac+bc) a b+ c+ 128. (2.10) 8 Art Discrete Appl. Math. 1 (2018) #P1.09 Now substitute (2.10) and (2.7) with e = 1 into the left-hand side of (2.8). When we expand the resulting expression, all the coefficients will be positive. Since the constant term is 8280, which is strictly positive, (2.8) is satisfied and so R3(Ha,b,c,1,1) > R3(Pa+b+c+4). Case 3: a, b, c, d 2, e = 0. In [8] we have = 4(a+b+c+d)2 11(ab+ac+ad+bc+bd+cd) + 3(a+b+c+d) + 137. (2.11) Denote D = 10 cd(a b)2(a+ b) + bd(a c)2(a+ c) + ad(b c)2(b+ c) + bc(a d)2(a+ d) + ac(b d)2(b+ d) + ab(c d)2(c+ d) . Observe that D 0. Now substitute (2.11) and (2.7) into the left-hand side of (2.8) and delete D. When we expand the resulting expression, all the coefficients will be pos- itive. Since the constant term is 828, which is strictly positive, (2.8) is satisfied and so R3(Ha,b,c,d,0) > R3(Pa+b+c+d+2). This completes the proof. References [1] F. Buckley, Mean distance in line graphs, in: F. Hoffman, K. B. Reid, R. C. Mullin and R. G. Stanton (eds.), Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Volume I, Utilitas Mathematica Publishing, Winnipeg, Manitoba, 1981 pp. 153–162, held at Louisiana State University, Baton Rouge, Louisiana, March 2 – 5, 1981. [2] N. Cohen, D. Dimitrov, R. Krakovski, R. Škrekovski and V. Vukašinović, On Wiener index of graphs and their line graphs, MATCH Commun. Math. Comput. Chem. 64 (2010), 683–698, http://match.pmf.kg.ac.rs/electronic_versions/ Match64/n3/match64n3_683-698.pdf. [3] A. A. Dobrynin and L. S. Mel’nikov, Wiener index of line graphs, in: I. Gutman and B. Furtula (eds.), Distance in Molecular Graphs – Theory, University of Kragujevac, Kragujevac, vol- ume 12 of Mathematical Chemistry Monographs, pp. 85–121, 2012, http://match.pmf. kg.ac.rs/mcm12.html. [4] J. K. Doyle and J. E. Graver, Mean distance in a graph, Discrete Math. 7 (1977), 147–154, doi:10.1016/0012-365x(77)90144-3. [5] R. C. Entringer, D. E. Jackson and D. A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976), 283–296, https://dml.cz/handle/10338.dmlcz/101401. [6] I. Gutman, Distance of line graphs, Graph Theory Notes N. Y. 31 (1996), 49–52. [7] I. Gutman and I. G. Zenkevich, Wiener index and vibrational energy, Z. Naturforsch. A 57 (2002), 824–828, http://www.znaturforsch.com/aa/v57a/s57a0824.pdf. [8] M. Knor, M. Mačaj, P. Potočnik and R. Škrekovski, Complete solution of equation W (L3(T )) = W (T ) for the Wiener index of iterated line graphs of trees, Discrete Appl. Math. 171 (2014), 90–103, doi:10.1016/j.dam.2014.02.007. [9] M. Knor, P. Potočnik and R. Škrekovski, On a conjecture about Wiener index in iterated line graphs of trees, Discrete Math. 312 (2012), 1094–1105, doi:10.1016/j.disc.2011.11.023. [10] M. Knor, P. Potočnik and R. Škrekovski, The Wiener index in iterated line graphs, Discrete Appl. Math. 160 (2012), 2234–2245, doi:10.1016/j.dam.2012.04.021. [11] M. Knor, P. Potočnik and R. Škrekovski, Wiener index of iterated line graphs of trees homeo- morphic to H, Discrete Math. 313 (2013), 1104–1111, doi:10.1016/j.disc.2013.02.005. K. Hriňáková et al.: On a conjecture about the ratio of Wiener index in iterated line graphs 9 [12] M. Knor, P. Potočnik and R. Škrekovski, Wiener index of iterated line graphs of trees homeomorphic to the claw K1,3, Ars Math. Contemp. 6 (2013), 211–219, https:// amc-journal.eu/index.php/amc/article/view/250. [13] M. Knor and R. Škrekovski, Wiener index of line graphs, in: M. Dehmer and F. Emmert-Streib (eds.), Quantitative Graph Theory: Mathematical Foundations and Applications, CRC Press, Boca Raton, FL, Discrete Mathematics and its Applications (Boca Raton), pp. 279–301, 2015. [14] M. Knor, R. Škrekovski and A. Tepeh, An inequality between the edge-Wiener index and the Wiener index of a graph, Appl. Math. Comput. 269 (2015), 714–721, doi:10.1016/j.amc.2015. 07.050. [15] M. Knor, R. Škrekovski and A. Tepeh, Mathematical aspects of Wiener index, Ars Math. Con- temp. 11 (2016), 327–352, https://amc-journal.eu/index.php/amc/article/ view/795. [16] J. Plesnı́k, On the sum of all distances in a graph or digraph, J. Graph Theory 8 (1984), 1–21, doi:10.1002/jgt.3190080102. [17] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), 17–20, doi:10.1021/ja01193a005. [18] K. Xu, M. Liu, K. Ch. Das, I. Gutman and B. Furtula, A survey on graphs ex- tremal with respect to distance-based topological indices, MATCH Commun. Math. Com- put. Chem. 71 (2014), 461–508, http://match.pmf.kg.ac.rs/electronic_ versions/Match71/n3/match71n3_461-508.pdf. ISSN 2590-9770 The Art of Discrete and Applied Mathematics 1 (2018) #P1.10 https://doi.org/10.26493/2590-9770.1258.c2b (Also available at http://adam-journal.eu) The number of independent sets in a connected graph and its complement Yumei Hu ⇤, Yarong Wei School of Mathematics, Tianjin University, Tianjin, PR China Received 29 March 2017, accepted 3 January 2018, published online 26 July 2018 Abstract For a connected graph G, the total number of independent vertex sets (including the empty vertex set) is denoted by i(G). In this paper, we consider Nordhaus-Gaddum-type inequalities for the number of independent sets of a connected graph with connected com- plement. First we define a transformation on a graph that increases i(G) and i(G). Next, we obtain the minimum and maximum value of i(G)+i(G), where graph G is a tree T with connected complement and a unicyclic graph G with connected complement, respectively. In each case, we characterize the extremal graphs. Finally, we establish an upper bound on the i(G) in terms of the Wiener polarity index. Keywords: Independent sets, connected complement, bounds, the Wiener polarity index, Nordhaus- Gaddum-type inequality. Math. Subj. Class.: 05C69, 05C30 1 Introduction Let G = (V (G), E(G)) be a simple connected graph of order n with vertex set V (G) and edge set E(G), denote by NG(u) the set of neighbors of a vertex u in G, and denote by G[S] the graph which is induced by vertex set S ✓ V (G). A double star Sp,q is obtained from Sp and Sq by connecting the center of Sp with that of Sq . A graph is unicyclic if and only if it is connected and has size equal to its order. Given a graph G, a k-independent set is a set of k vertices, no two of which are adjacent. Denote by i(G; k) the number of k-independent sets of G, k 1. It is both consistent and convenient to define i(G; 0) = 1. The family of the independent sets in G which contains the vertex sets U and S is denoted by IU,S(G), and let iU,S(G; k) be its cardinality. The ⇤The corresponding author, Yumei Hu, supported by NSFC NO. 11001196. E-mail addresses: huyumei@tju.edu.cn (Yumei Hu), math5025@163.com (Yarong Wei) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Art Discrete Appl. Math. 1 (2018) #P1.10 total number of independent vertex sets (including the empty vertex set) of a molecular graph G = (V,E), denoted by i(G), is defined as i(G) = X k0 i(G; k). In chemical literature, the number of the independent sets of graphs i(G) is referred to as the Merrifield-Simmons index. It is a valuable topological index introduced by the American chemists Richard E. Merrifield and Howard E. Simmons [12] in 1989. It is one of the topological indices whose mathematical characteristics has been extensively studied in a monograph [11, 20]. Its applicability for QSPR and QSAR was also examined to a much lesser extent. In [12] it has been shown that i(G) is correlated with boiling points. And, for the path Pn, i(Pn) is equal to the Fibonacci number Fn+1 [15]. The problem of counting the number of independent sets in a graph is NP-complete [16]. However, for certain types of graphs the problem of determining their number of indepen- dent subsets is polynomial. For instance, the number of independent sets in tree, uncyclic, and tricyclic graphs are calculated in [15, 14, 21], respectively. It is of significant interest to study the extremal graphs having maximal or minimal index. Zhu [20] characterized the extremal unicyclic graphs with a perfect matching which have maximal, second maximal Merrifield-Simmons index. In [17], S. Wagner and I. Gutman wrote a survey of results and techniques on the Hosoya index and Merrifield-Simmons index. Other recent results on the number of independent sets can be found in [2, 4, 3, 9]. The number of unordered vertices pairs that are at distance 3 in a graph G, denoted by Wp(G), is Wp(G) = |{(u, v) | dG(u, v) = 3, u, v 2 V (G)}|. It is also referred as the Wiener polarity index ([5, 7, 8]). Motivated by the result of [7], Hua et al. gave an upper bound on the Wiener polarity index in terms of the Hosoya index. We can find that, in a graph G, every pair of vertices at distance 3 corresponds to some 2-independent sets. There are also some relationships between the number of independent sets and the Wiener polarity index. The Nordhaus-Gaddum-type results are bounds of the sum or the product of a parameter for a graph and its complement. The name “Nordhaus-Gaddum-type” is given because Nordhaus and Gaddum [13] first found this type of inequality for the chromatic number of a graph and its complement in 1956. Since then, Nordhaus-Gaddum-type inequalities for many other graph invariants have been studied in a number of papers [1, 7, 10, 19]. We respectively research Nordhaus-Gaddum-type results for tree i(T ), unicyclic graph i(G) and connected graph i(G). In this paper, we consider Nordhaus-Gaddum-type inequalities for the number of inde- pendent sets of a connected graph with a connected complement. Firstly, in Section 2 we establish a transformation on graphs that increases i(G) and i(G). Secondly, in Section 3 and 4, we obtain the minimum and maximum value of i(G) + i(G), where graph G is a tree T with connected complement and a unicyclic graph G with connected complement, respectively. In each case, we characterize the extremal graphs. Finally, in Section 5 we es- tablish a lower bound on i(G) in terms of the Wiener polarity index. And, for a connected graph G with connected complement G, we obtain the minimum of i(G)+ i(G). Also, we pose a conjecture about which graph obtains the maximum value of i(G) + i(G). Other notation and terminology not defined here will conform to those in [18]. Y. Hu and Y. Wei: The number of independent sets in a connected graph and its complement 3 2 Preliminary Lemma 2.1 ([6]). Let G = (V,E) be a graph. (1) If uv 2 E(G), then i(G) = i(G uv) i(G (N [u] [N [v])); (2) If u 2 V (G), then i(G) = i(G u) + i(GN [u]); (3) If G1, G2, . . . , Gt are the components of the graph G, then i(G) = Q t j=1 i(Gj). Theorem 2.2. Let G be a simple graph and uv an edge of G such that NG(u)\NG(v) = ; and d(u), d(v) > 1. Let Gu,v denote the graph obtained from G by identifying vertex u and v (the new vertex is labeled as u) and attaching a pendent vertex v at u. Then (1) i(Gu,v) i(G) with equality if and only if G[NGu(v) [ NG(u) \ {v}] is not an empty graph; (2) i(Gu,v) i(G) with equality if and only if G[NGu(v) [ NG(u) \ {v}] is not an empty graph. Proof. For convenience, let G0 = Gu,v . By Lemma 2.1(1), for all non-negative integers k, we have i(G; k) = i(G u; k) + i(G uNG(u); k 1). (1) By i(G; k) = i(G u; k) + i(G uNG(u); k 1), we have i(G; k) = i(G v; k) + i(G v N G (v); k 1) = i(G v u; k) + i(G v uN Gv(u); k 1) + i(G v NG(v); k 1) and i(G0; k) = i(G0 v u; k) + i(G0 v uN G0v(u); k 1) + i(G0 v N G0(v); k 1). Obviously, G v u = G0 v u, N Gv(u) = NG0v(u) [ (NG(v) \ {u}), N G0(v) = NG(v) [ (NG(v) \ {u}), N G (v) = [N G (u) \ (NG(v) \ {u})] [ [NG(u) \ {v}], G0 v N G0(v) = G v NG(v) (NG(v) \ {u}), N G (v) \ (NG(v) \ {u}) = ;, and N G0v(u) \ (NG(v) \ {u}) = ;. So, i(G0; k) i(G; k) = i(G v u; k) + i(G v N G (v) (NG(v) \ {u}); k 1) + i(G v u [N G (u) \ (NG(v) \ {u})]; k 1) i(G v u; k) i(G v uN Gv(u); k 1) i(G v NG(v); k 1) = iNG(v)\{u}(G v u [NGv(u) \ (NG(v) \ {u})]; k 1) iNG(v)\{u}(G v NG(v); k 1) 4 Art Discrete Appl. Math. 1 (2018) #P1.10 = iNG(v)\{u}(G v u [NGv(u) \ (NG(v) \ {u})]; k 1) iNG(v)\{u}(G v uNG(v); k 1) iNG(v)\{u}(G v uNG(v)NGvNG(v)(u); k 2) = iNG(v)\{u},NG(u)\{v}(G v u [NGv(u) \ (NG(v) \ {u})]; k 1) 0. Obviously, if iNG(v)\{u},NG(u)\{v} G v u [N Gv(u) \ (NG(v) \ {u})]; k 1 = 0, G[NGu(v) [ NG(u) \ {v}] is not an empty graph. Conversely, if graph G[NGu(v) [ NG(u) \ {v}] is not an empty graph, iNG(v)\{u},NG(u)\{v} G v u [N Gv(u) \ (NG(v) \ {u})]; k 1 = 0. (2) By i(G; k) = i(G u; k) + i(G uNG(u); k 1), we can similarly get: i(G; k) = i(G u; k) + i(G uNG(u); k 1) = i(G u v; k) + i(G u v NGu(v); k 1) + i(G uNG(u); k 1) and i(G0; k) = i(G0 u v; k) + i(G0 u v NG0u(v); k 1) + i(G0 uNG0(u); k 1). Obviously, G u v = G0 u v, NG0u(v) = ;, NG(u) \NGu(v) = ;, and G0 uNG0(u) = G uNG(u)NGu(v). So, i(G0; k) i(G; k) = i(G u v; k) + i(G uNG(u)NGu(v); k 1) + i(G u v; k 1) i(G u v NGu(v); k 1) i(G u v; k) i(G uNG(u); k 1) = iNGu(v)(G u v; k 1) iNGu(v)(G uNG(u); k 1) = iNGu(v),NG(u)\{v}(G u v; k 1) 0. Obviously, if iNGu(v),NG(u)\{v}(G u v; k 1) = 0, G[NGu(v) [ NG(u) \ {v}] is not an empty graph. Conversely, if graph G[NGu(v) [ NG(u) \ {v}] is not an empty graph, iNGu(v),NG(u)\{v}(G u v; k 1) = 0. Y. Hu and Y. Wei: The number of independent sets in a connected graph and its complement 5 3 The Nordhaus-Gaddum-type inequality for trees In this section, we consider a tree T with connected complement T , then we obtain the minimum and maximum value of i(T ) + i(T ) and characterize the extremal graph. Lemma 3.1 ([15]). The star Sn has the maximal Merrifield-Simmons index for all trees with n vertices. And, the path Pn has the minimal Merrifield-Simmons index for all trees with n vertices. For the proof, we give an equality involving i(T ) + i(T ) as follows. Lemma 3.2. Let T be a tree of order n with connected complement T . Then i(T ) + i(T ) = 2n+ i(T ). Proof. For connected complement T and all non-negative integers k 3, it is easy to verify i(T ; k) = 0 and i(T ; 2) = |E(T )| = n 1. Therefore i(T ) + i(T ) = i(T ) + 1 + n+ i(T ; 2) = 2n+ i(T ). Now we give the Nordhaus-Gaddum-type inequality of a tree for i(T ). Theorem 3.3. Let T be a tree of order n with connected complement T , then i(T ) + i(T ) 2n+ Fn+1 with equality if and only if T ⇠= Pn, where Fn+1 is the Fibonacci number. Proof. By Lemma 3.1 and Lemma 3.2 graph which reaches the minimum value of i(T ) + i(T ). And, i(Pn) is equal to the Fibonacci number Fn+1, then i(T )+i(T ) 2n+Fn+1. Theorem 3.4. Let T be a tree of order n with connected complement T , then i(T ) + i(T )  2 + 2n+ 2n3 + 2n2 with equality if and only if T ⇠= S2,n2. Proof. If T and T are connected graphs, then the star Sn is not the extremal graph which reaches the maximum value of i(T ) + i(T ). So we assume D(T ) 3. Let P = v0v1 . . . vD(T ) be a diametrical path of tree T . By Theorem 2.2, we have i(TD(T )1,D(T )2) + i(TD(T )1,D(T )2) > i(T ) + i(T ). Obviously, graph TD(T )1,D(T )2 is a tree of order n. Therefore, for the tree of order n with connected complement, by shortening the dia- metrical path of a tree, we can get the extremal graph the double star Sp,q which reaches the maximal value of i(T ) + i(T ). For the double star Sp,q of order n, we have i(Sp,q) + i(Sp,q) = 2 + 2n+ n2X i=2 ✓ n 2 i ◆ + p1X i=1 ✓ p 1 i ◆ + q1X i=1 ✓ q 1 i ◆ + p+ q 1 = 2n+ 2n2 + 2p1 + 2q1  2 + 2n+ 2n3 + 2n2 with equality if and only if p = n2 or q = n2. So i(T )+i(T )  2+2n+2n3+2n2 with equality if and only if T ⇠= S2,n2. 6 Art Discrete Appl. Math. 1 (2018) #P1.10 4 The Nordhaus-Gaddum-type inequality for unicyclic graphs In this section, we consider a unicyclic graph G of order n with connected complement G, then we obtain the minimum and maximum value of i(G) + i(G) and characterize the extremal graph. Obviously, if n < 5, any complement G is not connected. We need to consider the case when n 5. Lemma 4.1 ([14]). If G is a unicyclic graph of order n, then (1) i(G) Fn1+Fn+1 and equality occurs if and only if G ⇠= Cn or G ⇠= Ln,3, where Ln,3 is the unicyclic graph of order n obtained from the two vertex disjoint graphs C3 and Pn3 by adding an edge joining a vertex of C3 to an endvertex of Pn3. (2) i(G)  3⇥ 2n3 + 1 and equality holds if and only if G is a 4-cycle or G ⇠= Hn,3, where Hn,3 is the unicyclic graph of order n constructed by attaching n 3 leaves to one vertex on a cycle of length 3. For the proof, we give an equality about i(G) + i(G) as follows. Lemma 4.2. Let G be a unicyclic graph of order n 5 with connected complement G, then i(G) + i(G) = 1 + 2n+ i(G; 3) + i(G). Proof. For connected complement G and all non-negative integers k 4, it is easy to verify i(G; k) = 0 and i(G; 2) = |E(G)| = n. Therefore i(G) + i(G) = i(G) + 1 + n+ i(G; 2) + i(G; 3) = 1 + 2n+ i(G; 3) + i(G). Now we give the Nordhaus-Gaddum-type inequality of a unicyclic graph for i(G). Theorem 4.3. Let G be a unicyclic graph of order n 5 with connected complement G, then i(G) + i(G) 1 + 2n+ Fn1 + Fn+1 with equality if and only if G ⇠= Cn, where Fn+1 is the Fibonacci number. Proof. Obviously, i(Ln,3; 3) = 1 > i(Cn; 3) = 0, and the complement of graph Cn is a connected graph. Then by Lemma 4.1(1) and Lemma 4.2, we have i(Ln,3) = i(Cn) and i(G) + i(G) = 1 + 2n+ i(G; 3) + i(G) 1 + 2n+ i(Cn; 3) + i(Cn) = 1 + 2n+ Fn1 + Fn+1. In order to formulate our results, some graphs need to be defined. Let Ox1,x2,x3 denote a unicyclic graph on n vertices created from a cycle C3 = v1v2v3 by attaching xi (i = 1, 2, 3) pendent vertices to vi such that x1 + x2 + x3 + 3 = n and x1 x2 x3, x2 1. Let Uy1,y2 denote a unicyclic graph on n vertices created from a cycle C3 = v1v2v3 by attaching y1 pendent vertices u1, u2, . . . , uy1 to v1 and attaching y2 pendent vertices to u1 such that y1 + y2 + 3 = n and y1 1, y2 2. Theorem 4.4. Let G be a unicyclic graph of order n 5 with connected complement G, then i(G) + i(G)  4 + 2n+ 2n4 + 2n2 with equality if and only if G ⇠= On4,1,0. Y. Hu and Y. Wei: The number of independent sets in a connected graph and its complement 7 Proof. If the cycle in G is of length greater than three, then by applying the transformation in Theorem 2.2 to the cycle, there is a unicyclic graph L1 with a triangle such that i(L1) + i(L1) > i(G) + i(G). Let H denote the set of all unicyclic graphs H with a triangle. Then for all H 2 H by Lemma 4.2 we have i(H) + i(H) = 1 + 2n + i(H; 3) + i(H) = 2 + 2n + i(H). The maximum value of i(H)+i(H) is equal to the maximum value of i(H). By Lemma 4.1(2), we know that the graph Hn,3 is the extremal graph which obtains the maximum value i(H), but the graph Hn,3 is not connected. So we calculate the second maximum value of i(H). Case 1: There is a vertex in H with distance at least two to the 3-cycle. By the transformation in Theorem 2.2, we get there is a graph L2 with i(L2) > i(H), where L2 ⇠= Ox1,x2,x3 or L2 ⇠= Uy1,y2 . i(Uy1,y2) = i(Uy1,y2 v3) + i(Uy1,y2 NUy1,y2 [v3]) = i(Sy1+1,y2+1) + i(Ky11 [ Sy2+1) = 3⇥ 2n4 + 2y2 + 3⇥ 2y11  3⇥ 2n4 + 2 + 3⇥ 2n5 < 2 + 2n4 + 2n2 = i(On4,1,0) Case 2: G ⇠= Ox1,x2,x3 i(Ox1,x2,x3) = i(Ox1,x2,x3 v3) + i(Ox1,x2,x3 NOx1,x2,x3 [vv3 ]) = i(Sx1+1,x2+2 [Kx3) + i(Kx1+x2) = 2n3 + 2x1+x3 + 2x2+x3 + 2n3x3  2n3 + 2x1+x3 + 2x2 + 2n3 = i(Ox1+x3,x2,0)  2n3 + 2n4 + 2 + 2n3 = i(On4,1,0) with equality if and only if x3 = 0 and x2 = 1. Obviously, the graph On4,1,0 is connected. So i(G) + i(G)  i(H) + i(H)  4 + 2n + 2n4 + 2n2 with equality if and only if G ⇠= On4,1,0. 5 The Nordhaus-Gaddum-type inequality for connected graphs In this section, we obtain a lower bound on i(G) in terms of the Wiener polarity index. And, for a connected graph G with connected complement G, we obtain a minimum value of i(G) + i(G) and characterize the extremal graph. Also, we pose a conjecture about which graph gets the maximum value of i(G) + i(G). Lemma 5.1 ([7]). Let G be a connected graph with connected complement G, then Wp(G) +Wp(G) D(G) +D(G) 4. Moreover, equality holds if and only if G ⇠= Pn or G ⇠= G⇤⇤ or D(G) = D(G) = 2. The graph G⇤⇤ of order n 5 is obtained from a path P4 by joining each vertex of Hn4 to each internal vertex of the path P4 such that V (G⇤⇤)\V (P4) = V (Hn4), where Hn4 is any graph of order n 4. 8 Art Discrete Appl. Math. 1 (2018) #P1.10 In order to get the lower bound on the i(G) + i(G), we give a lower bound on the i(G) in terms of the Wiener polarity index. Lemma 5.2. Let G be a connected graph of order n and D(G) 2. Then i(G) 2 + n+ 2Wp(G) with equality if and only if G ⇠= Gn or G ⇠= B2,n2, where Gn = Kn e, e 2 E(Kn), B2,n2 is a graph on n 3 vertices obtained from P2 and Kn2 by coinciding any vertex of P2 with that of Kn2. Proof. If D(G) = 2, then Wp(G) = 0. Let P = uxy be a diametrical path, then {u, y} is a 2-independent set of G. Therefore i(G) i(G; 0) + i(G; 1) + i(G; 2) 1 + n+ 1 = 2 + n+ 2Wp(G) follows readily. Suppose that equality is attained. Then G has only one 2-independent set and no k-independent set, where k 3. Also, D(G) = 2. Then, we have G ⇠= Gn. Conversely, if G ⇠= Gn, then the equality is attained. For the case D(G) 3: Suppose that u and v are a pair of vertices in G such that dG(u, v) = 3. Let uxyv be a path of length 3 connecting u and v in G. Then {u, y}, {u, v} and {x, v} are 2-independent sets of G. Therefore, every pair of vertices at distance 3 corresponds to three 2-independent sets in G. Moreover, for any two paths connecting distinct pair vertices at distance 3, they correspond to two different 2-independent sets and one same 2-independent set, otherwise they correspond to three different 2-independent sets. From this it follows that i(G; 2) 2Wp(G) + 1. Therefore, by the definition of Merrifield-Simmons index, i(G) i(G; 0) + i(G; 1) + i(G; 2) 1 + n+ 2Wp(G) + 1 = 2 + n+Wp(G), (2) follows readily. Now, we check the equality condition. If i(G; 2) = 2Wp(G) + 1, by analysis, then any two paths of distinct pair vertices at distance 3 correspond to two different 2-independent sets and one same 2-independent set. If i(G; 3) = 0, D(G) = 3. So G ⇠= B2,n2. Conversely, if G ⇠= B2,n2, then we clearly have i(G; 2) = 2Wp(G)+1 and i(G; 3) = 0. So, the equality is attained if and only if G ⇠= B2,n2. Theorem 5.3. Let G be a connected graph with connected complement G, then i(G) + i(G) 2n+ 2D(G) + 2D(G) 4 with equality if and only if G ⇠= P4. Proof. By Lemma 5.1 and Lemma 5.2, the result is obvious. Conjecture 5.4. Let G be a connected graph with connected complement G, then i(G) + i(G)  2 + 2n+ 2n3 + 2n2 with equality if and only if G ⇠= S2,n2. Y. Hu and Y. Wei: The number of independent sets in a connected graph and its complement 9 For a connected graph G with connected complement G, it is difficult to get the value of max{i(G) + i(G)}. For n  5, by enumeration and calculation, we can find max{i(G) + i(G)} = max{i(T ) + i(T )} = i(S2,n2) + i(S2,n2). If we do not consider the connec- tivity of the graph, we can get: Theorem 5.5. Let G be a simple graph of order n. If we do not consider the connectivity of the graph, then i(G) + i(G)  1 + n+ 2n with equality if and only if G ⇠= Kn or G ⇠= Kn. Proof. Let k,m 2 N. Without loss of generality, we assume that ↵(G) > ↵(G). Every pair of vertices are not a 2-independent set of G, which compose of a 2-independent set of G. Moreover, for any two vertices which do not compose of a 2-independent set of G, they compose of a 2-independent set of G. Then, we have i(G; 2) + i(G; 2) = C2 n . Suppose i(G; 3) = k. Since every three vertices which are a 3-independent set of G are not a 3-independent set of G, we have i(G; 3)  C3 n k. Therefore, we have i(G) + i(G)  2 + 2n+ C2 n + C3 n + . . .+ C↵(G) n = 1 + n+ 2n nX i=↵(G)+1 Ci n . (5.1) Now, we check the equality condition in (1). If i(G;m) = Cm n i(G;m), then for any m vertices which are not an independent set of G, they are an m-independent set of G. Then, G is the empty graph, and G ⇠= Kn. By the definition of G and G, we have i(G) + i(G)  1 + n+ 2n. Obviously, for connected graph G with a connected complement, 1+n+2n is an upper bound on the maximum value of i(G)+i(G). And, the lower bound on the maximum value of i(G) + i(G) is i(S2,n2) + i(S2,n2). The difference between the upper bound and the lower bound is 5 · 2n3 n 1. 6 Conclusions In this paper, we firstly establish a transformation on a simple graph that increases i(G) and i(G). Secondly, we prove the path Pn and the double star S2,n2 are the extremal graphs which respectively reach the minimum and maximum value of i(T ) + i(T ). Then, for unicyclic graphs G, we get that the cycle Cn and the graph On4,1,0 are the extremal graphs which respectively reach the minimum and maximum value of i(G)+i(G). Finally, for connected graphs G, we find i(G) 2 + n + 2Wp(G) with equality if and only if G ⇠= Gn or G ⇠= B2,n2. Then we obtain i(G) + i(G) 2n+ 2D(G) + 2D(G) 4 with equality if and only if G ⇠= P4. Also, we conjecture that the extremal graph which reaches the maximum value of i(G) + i(G) is S2,n2. Which graph gives the maximum value on i(G) + i(G) remains an open problem. References [1] M. Aouchiche and P. Hansen, A survey of Nordhaus-Gaddum type relations, Discrete Appl. Math. 161 (2013), 466–546, doi:10.1016/j.dam.2011.12.018. 10 Art Discrete Appl. Math. 1 (2018) #P1.10 [2] H. Chen, R. Wu, G. Huang and H. Deng, Independent sets on the Towers of Hanoi graphs, Ars Math. Contemp. 12 (2017), 247–260, https://amc-journal.eu/index.php/amc/ article/view/783. [3] S. Debroni, E. Delisle, W. Myrvold, A. Sethi, J. Whitney, J. Woodcock, P. W. Fowler, B. de La Vaissière and M. Deza, Maximum independent sets of the 120-cell and other regular polytopes, Ars Math. Contemp. 6 (2013), 197–210, https://amc-journal.eu/index. php/amc/article/view/170. [4] H. Deng and S. Chen, The extremal unicyclic graphs with respect to Hosoya index and Merrifield-Simmons index, MATCH Commun. Math. Comput. Chem. 59 (2008), 171– 190, http://match.pmf.kg.ac.rs/electronic_versions/Match59/n1/ match59n1_171-190.pdf. [5] W. Du, X. Li and Y. Shi, Algorithms and extremal problem on Wiener polarity index, MATCH Commun. Math. Comput. Chem. 62 (2009), 235–244, http://match.pmf.kg.ac.rs/ electronic_versions/Match62/n1/match62n1_235-244.pdf. [6] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin, 1986, doi:10.1007/978-3-642-70982-1. [7] H. Hua and K. C. Das, On the Wiener polarity index of graphs, Appl. Math. Comput. 280 (2016), 162–167, doi:10.1016/j.amc.2016.01.043. [8] H. Lei, T. Li, Y. Shi and H. Wang, Wiener polarity index and its generalization in trees, MATCH Commun. Math. Comput. Chem. 78 (2017), 199–212, http://match.pmf.kg.ac.rs/ electronic_versions/Match78/n1/match78n1_199-212.pdf. [9] H.-H. Li, Q.-Q. Wu and I. Gutman, On ordering of complements of graphs with respect to matching numbers, Appl. Math. Comput. 282 (2016), 167–174, doi:10.1016/j.amc.2016.02. 004. [10] X. Li and Y. Mao, Nordhaus-Gaddum-type results for the generalized edge-connectivity of graphs, Discrete Appl. Math. 185 (2015), 102–112, doi:10.1016/j.dam.2014.12.009. [11] X. Li, H. Zhao and I. Gutman, On the Merrifield-Simmons index of trees, MATCH Com- mun. Math. Comput. Chem. 54 (2005), 389–402, http://match.pmf.kg.ac.rs/ electronic_versions/Match54/n2/match54n2_389-402.pdf. [12] R. E. Merrifield and H. E. Simmons, Topological Methods in Chemistry, A Wiley-Interscience Publication, Wiley, New York, 1989. [13] E. A. Nordhaus and J. W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956), 175–177, doi:10.2307/2306658. [14] A. S. Pedersen and P. D. Vestergaard, The number of independent sets in unicyclic graphs, Discrete Appl. Math. 152 (2005), 246–256, doi:10.1016/j.dam.2005.04.002. [15] H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982), 16–21, https://www.fq.math.ca/Scanned/20-1/prodinger.pdf. [16] D. Roth, On the hardness of approximate reasoning, Artif. Intell. 82 (1996), 273–302, doi: 10.1016/0004-3702(94)00092-1. [17] S. Wagner and I. Gutman, Maxima and minima of the Hosoya index and the Merrifield- Simmons index: a survey of results and techniques, Acta Appl. Math. 112 (2010), 323–346, doi:10.1007/s10440-010-9575-5. [18] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, New Jersey, 2nd edition, 2001. [19] Y. Zhang and Y. Hu, The Nordhaus-Gaddum-type inequality for the Wiener polarity index, Appl. Math. Comput. 273 (2016), 880–884, doi:10.1016/j.amc.2015.10.045. Y. Hu and Y. Wei: The number of independent sets in a connected graph and its complement 11 [20] Z. Zhu, The extremal unicyclic graphs with perfect matching with respect to Hosoya in- dex and Merrifield-Simmons index, Ars Combin. 124 (2016), 277–287, http://www. combinatorialmath.ca/arscombinatoria/vol124.html. [21] Z. Zhu, S. Li and L. Tan, Tricyclic graphs with maximum Merrifield-Simmons index, Discrete Appl. Math. 158 (2010), 204–212, doi:10.1016/j.dam.2009.09.001. Norman W. Johnson (12 November 1930 to 13 July 2017) Norman W. Johnson was born on November 12, 1930 in Chicago, where his father had a bookstore and ran a local newspaper. He attended Carleton College, graduating in 1953. He did alternative service as a conscientious objector then went on to earn a Master’s degree from the University of Pittsburgh. He then went to the University of Toronto to work with H. S. M. Coxeter in geometry. After receiving his PhD in 1966 he accepted a position in the Mathematics Department of Wheaton College in Massachusetts and taught there until his retirement in 1998. He is known for the “Johnson Solids,” the ninety-two non-uniform con- vex solids with regular faces that he identified in a 1966 article [1] and speculated was the complete set. He also published a number of other articles on various aspects of polytopes. He died on July 13, 2017, but his completed book, Geometries and Transformations [2], is forthcoming from Cambridge University Press. His nearly-completed work on uniform polytopes, the subject of his dissertation, will be appearing. Asia Ivić Weiss and Eva Marie Stehle References [1] N. W. Johnson, Convex polyhedra with regular faces, Canad. J. Math. 18 (1966), 169–200, doi: 10.4153/cjm-1966-021-8. [2] N. W. Johnson, Geometries and Transformations, Cambridge University Press, Cambridge, 2017. Professor Wilfried Imrich awarded honorary doctorate at the University of Maribor On January 31, 2018, Prof. Emer. Dr. Wilfried Imrich from the Montanuniversität Leoben, Austria, became a Honorary Doctor of the University of Maribor. The title was awarded to him for his scientific achievements and contributions to the development of the University of Maribor. The university awards this title since 1979, Wilfried Imrich is the first math- ematician to receive this prestigious title. Moreover, he is the first foreign mathematician with the honorary doctor title at a Slovenian university. The collaboration between Wilfried Imrich and the Slovenian graph theory school started when in the 1980s he established together with Tomo Pisanski the Leoben-Ljubljana seminar, which is still going on. The rest is then history. As a coincidence, the 30th Ljubljana-Leoben Graph Theory Seminar that happened in September 2017, took place for the first time in Maribor. In the last two decades, Wilfried was a frequent participant of the Seminar on discrete mathematics that is held at the Faculty of Natural Sciences and Mathematics in Maribor. He has written three books and close to fifty papers with a dozen co-authors from Maribor. The fact that at the present 16 academic descendants of Prof. Imrich have positions at the University of Maribor indicates that the award was more than deserved. Boštjan Brešar and Sandi Klavžar Special Issue of ADAM on Symmetries of Graphs and Networks – Call for Papers This is a call for submission of papers for a special issue of the journal The Art of Discrete and Applied Mathematics (ADAM), on topics presented or related to talks given at the TSIMF workshop on ‘Symmetries of Graphs and Networks’ held at Sanya (China) in January 2018. The Sanya workshop added to the series of conferences and workshops on symmetries of graphs and networks initiated at BIRS (Canada) in 2008 and progressed in Slovenia every two years from 2010 to 2016. The Art of Discrete and Applied Mathematics (ADAM) is a modern, dynamic, platinum open access, electronic journal that publishes high-quality articles in contemporary discrete and applied mathematics (including pure and applied graph theory and combinatorics), with no costs to authors or readers. This special issue, however, will be also available in printed form for purchase. Papers should be submitted by 31 December 2018, via the ADAM website https: //adam-journal.eu/index.php/ADAM. A template and style file for submissions can be downloaded from that website, or obtained from one of the guest editors on request. The ideal length of papers is 5 to 15 pages, but longer or shorter papers will certainly be considered. Papers that are accepted will appear on-line soon after acceptance, and papers that are not processed in time for the special issue may still be accepted and published in a subsequent regular issue of ADAM. Marston Conder and Yan-Quan Feng Guest Editors