/^creative ^commor ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 341-352 Linking Rings Structures and tetravalent semisymmetric graphs PrimoZ Potocnik Faculty of Mathematics and Physics, University of Ljubljana Jadranska 19, Ljubljana, Slovenia IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia Stephen E. Wilson Department of Mathematics and Statistics, Northern Arizona University Box 5717, Flagstaff, AZ 86011, USA Received 27 February 2012, accepted 27 October 2013, published online 7 August 2013 Abstract In this paper, we introduce LR structures, a new and interesting form of symmetry in graphs. LR structures are motivated by the search for semisymmetric graphs of degree 4. We show that all semisymmetric graphs of girth and degree 4 can be constructed in a simple way from LR structures. We then show several ways in which LR structures can be constructed or found. Keywords: Graph, automorphism group, symmetry, locally arc-transitive graph, semisymmetric graph, cycle structure, linking rings structure. Math. Subj. Class.: 20B25, 05E18 1 Introduction A regular graph r is semisymmetric provided that its symmetry group is transitive on edges but not on vertices. The search for semisymmetric graphs has an intense 40-year history, beginning with Harary and Dauber's paper [9]. This unpublished work was seen by Folk-man, and motivated his paper [5], which gives the smallest such graph, on 20 vertices. The earliest known semisymmetric graph, due to Marion Gray in the 1930's, shown in early dittoed versions of Foster's Census, was rediscovered by Bouwer, described in [1] and generalized in [2]. Progress on aspects of semisymmetric graphs was made in papers published E-mail addresses: primoz.potocnik@fmf.uni-lj.si (PrimoZ Potočnik), stephen.wilson@nau.edu (Stephen E. Wilson) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 340 Ars Math. Contemp. 7 (2014) 337-340 since 1980 (see for example [4, 8, 10, 11,12,17,18]). Recently, considerable attention was given to semisymmetric graphs of valence 3 [13, 14, 15], and a list of all such graphs with no more than 768 vertices was compiled in [3]. This paper continues work from [20], where the present authors began an investigation of tetravalent edge-transitive graphs, by considering those of girth at most 4. Such a graph is either worthy or unworthy, depending on whether the neighborhoods of all pairs of distinct vertices are distinct or not. Lemma 4.3 of [20] disposes of the unworthy case by showing that every unworthy tetravalent semisymmetric graph has order 4n for some integer n, and arises as the "subdivided double" of a dart-transitive tetravalent graph of order n. Note that the smallest semisymmetric graph, the Folkman graph on 20 vertices shown in Figure 1, is unworthy, and it can be obtained as a subdivided double of the complete graph K5. Similarly, it was shown in [17, Theorem 5.2] that every semisymmetric graph on 4p vertices, for p > 7 a prime, is unworthy. Figure 1: The Folkman graph On the other hand, [20, Theorem 5.1(5, b)] shows that every worthy tetravalent semi-symmetric graph of girth 4 and order 2n is isomorphic to a graph obtained, via the "partial line graph construction", from a certain kind of cycle decomposition of a vertex-transitive tetravalent graph of order n. A precise statement of this result can be found in Section 5. The purpose of the present paper is to study worthy tetravalent semisymmetric graphs of girth 4, and the cycle decompositions from which they arise, in more detail. 2 Preliminaries A standard notation for graphs is used throughout the paper: if r is a graph, then V(T) is its vertex set. An edge in r is an unordered pair from V(T) (so r is a simple graph), and the set of edges is E(T). An ordered pair of adjacent vertices will be called a dart (or also, an arc). In this paper, all graphs are finite and connected. Bipartition sets in bipartite graphs are often referred to as colors, black and white, of the vertices. A symmetry or automorphism of a graph r is a permutation of its vertices which preserves adjacency. The symmetries of r form a group, Aut(T), under composition. A graph is said to be vertex-, edge-, or dart-transitive if its automorphism group acts transitively on its vertices, edges, or darts, respectively. Dart-transitive graphs are also called symmetric or arc-transitive. P. Potocnik and S. E. Wilson: Linking Rings Structures and tetravalent semisymmetric. 343 A graph r is semisymmetric provided that it is connected, regular, and Aut(T) is transitive on edges but not on vertices. It follows easily (see [5] for example) that such a graph r is bipartite, and that Aut(T) is transitive on each of the two color classes of vertices. If for every vertex v of a regular graph r, the stabilizer Aut(T)v acts transitively on the neighbors of v, then r is said to be locally dart-transitive. It is well known that a locally dart-transitive graph is edge-transitive (recall that all the graphs are assumed to be connected), and that it is either dart-transitive or semisymmetric (depending on whether Aut(r) is transitive on the vertices or not). 3 Cycle decompositions Let A be a (connected) regular tetravalent graph and C a partition of E(A) into cycles (that is, sets of edges which induce in A connected subgraphs of valence 2). We shall call such a pair (A, C) a cycle decomposition. For the rest of this section, let (A, C) denote a cycle decomposition. Two edges of A will be called opposite at vertex v, if they are both incident with v and belong to the same element of C. The partial line graph of a cycle decomposition (A, C) is the graph P(A, C) whose vertices are edges of A, and two edges of A are adjacent in P(A, C) whenever they share a vertex in A and are not opposite at that vertex. The left hand side of Figure 2 shows a tetravalent graph A and a cycle decomposition C with cycles indicated by a-a, b-b, etc; and the right hand side of Figure 2 shows the partial line graph of (A, C). Figure 2: A cycle decomposition and its partial line graph An isomorphism between two cycle decompositions (Ai, Ci) and (A2, C2) is an isomorphism f of A1 onto A2 for which f (C1) = C2. An automorphism is an isomorphism from a cycle decomposition to itself. The group of automorphisms of (A, C) will be denoted by Aut(A, C). A cycle decomposition (A, C) is said to be flexible provided that for every vertex v and each edge e containing v, there is a symmetry in Aut(A, C) which fixes pointwise the cycle D G C containing e and interchanges the other two neighbors of v. The edges joining v to those neighbors are in some other cycle C of C, and the symmetry is called a C-swapper at v. A cycle decomposition (A, C) is called bipartite if C can be partitioned into two subsets G and R so that each vertex of A meets one cycle from G and one from R. Especially in constructions, we will refer to the edges of the cycles in G and those in R as green and red, 340 Ars Math. Contemp. 7 (2014) 337-340 respectively. The largest subgroup of Aut(A, C) preserving each of the sets G and R will be denoted by Aut+(A, C), and we will think of it as the color-preserving group of (A, C). Note that in a bipartite cycle decomposition, an element of Aut(A, C) either preserves each of the sets G and R setwise, and is thus contained in Aut+(A, C), or interchanges the sets G and R. (In particular, in a bipartite flexible structure, swappers belong to Aut+(A,C).) This shows that the index of Aut+(A,C) in Aut(A,C) is at most 2. If this index is 2, then we say that (A, C) is self-dual. Thus (A, C) is self-dual if and only if there is a e Aut(A, C) such that Go = R and Ra = G. We can now define the central notion of this paper: 4 LR Structures Definition 4.1. A cycle decomposition (A, C) is called a linking ring structure (or briefly, an LR structure) provided that it is bipartite, flexible and that Aut+(A, C) acts transtively on V(A). If (A, C) is an LR structure, then it follows from the definition that G = Aut+(A, C) acts transitively on red darts, as well as on green darts, in such a way that the permutation group G^(v), induced by the vertex-stabiliser Gv on the neighbourhood A(v), is intransitive and is in fact permutation isomorphic to the permutation group ((1, 2), (3,4)} < S4 (we shall call the permutation group ((1, 2), (3,4)} the intransitive Klein 4-group). Conversely, if a connected tetravalent graph A admits a vertex- but not edge-transitive group of automorphisms G such that G^(v) is permutation isomorphic to the intransitive Klein 4-group, then the two edge-orbits of G form the sets G and R of an LR structure (A, C) for which Aut+(A, C) contains G. This shows that, in the group-theoretical language, an LR structure could be equivalently defined as a transitive permutation group G admitting two self-paired orbitals A1 and A2 of degree 2 such that G^(v) is permutation isomorphic to the intransitive Klein 4-group. Since Aut+ (A, C) acts transitively on G and on R, it follows that all cycles in G must have the same length, say p, and all cycles in R must be of the same length q. We then say that the LR structure (A, C) is of type {p, q}. For a self-dual structure, of course, p = q. n 4 5 6 0 0 0 0 0 o0 3 O- 2o- 1o- ■«3 ■02 ■«31 , 4_5 6_ J □ □ , _4 5_6 n □ 13 :□ □ n 0O. .00 0o o0 0 0 Figure 3: Three LR structures on the 4-dimensional cube Q4, presented as a toroidal map {4, 4}4,0. The labels indicate the identifications of vertices and edges. Note that a tetravalent graph can admit more than one LR structure. For example, the 4-dimensional cube Q4 (see Figure 3) has three distinct bi-colorings of edges, each of which P. Potocnik and S. E. Wilson: Linking Rings Structures and tetravalent semisymmetric. 345 makes it into a self-dual LR structure of type {4,4}. The three structures are isomorphic. This leads us to pose the following open question: Question 1: Does there exist a graph admitting non-isomorphic LR structures? 5 Suitable LR Structures Let us now describe the relationship between LR structures and tetravalent edge-transitive graphs. If (A, C) is a cycle decomposition, then a cycle C in A is said to be alternating if no two consecutive edges of C belong to the same element of C. In [20], a cycle decomposition (A, C) was called smooth, provided that A contained no alternating 4-cycles, and no 3-cycles except possibly those contained in C. Note that the latter condition on 3-cycles is automatically satisfied for an LR structure: if (A, C) is an LR structure, and if bcd is a triangle in A, but not in C, then two of the three edges are the same color; say bc and cd are red, while bd is green. Then a swapper at b which fixes the red cycle containing bcd would interchange the two green neighbors at b. But one of these is d, which is left fixed by this symmetry. This is a contradiction, and so no such triangle can exist. Definition 5.1. An LR structure which is not self-dual and contains no alternating 4-cycles is called suitable. It is not difficult to see that a cycle decomposition (A, C) is a suitable LR structure if and only if it satisfies the conditions stated in [20, Theorem 5.1(5, b)]. This theorem, combined with Lemma 6.3, which will be proved in the following section, implies the following correspondence, which explains the relevance of LR structures to the study of tetravalent edge-transitive graphs. Theorem 5.2. The partial line graph construction P induces a bijective correspondence between the set of LR structures and the set of worthy bipartite locally dart-transitive tetravalent graphs of girth 4. If (A, C) is an LR structure, then P(A, C) is semisymmetric if and only if (A, C) is suitable; and is arc-transitive otherwise. If (A, C) contains an alternating 4-cycles, then P(A, C) is a skeleton of an arc-transitive map of type {4,4} on the torus. Remark. If a graph A has a cycle decomposition C for which (A, C) is a suitable LR structure, it is conceivable that it might allow a different cycle decomposition C for which (A, C ') is an LR structure. We know of no such example, and we conjecture that none exist. Proving that the conjecture holds would allow us to simplify our notation, and talk about an LR graph A. We will discuss this conjecture further in our final section. Every new LR structure gives a new edge-transitive graph, and so we are interested in finding and creating LR structures. The aim of this paper is to provide constructions of LR structures (and thus of bipartite tetravalent edge-transitive graphs), which show how varied the LR structures can be. 6 Basic constructions We begin our investigation of LR structures by presenting some basic contructions. 6.1 LR Structures on the wreath graphs The wreath graph W(n, 2) is defined to be the graph on 2n vertices Wj, vj, for i e Zn, with all edges from vertices of subscript i to those of subscript i + 1. Alternatively, W (n, 2) 340 Ars Math. Contemp. 7 (2014) 337-340 can be defined as the lexicographic product of the cycle Cn with the edgeless graph on two vertices. Construction 6.1. If n is even, let A be the wreath graph W(n, 2) and let C be the set of 4-cycles of W(n, 2) of the form [u.j, ui+1,vi, vi+1 ]. Color such a 4-cycle red if i is even and green if i is odd. Then (A, C) is clearly a self-dual LR-structure of type {4,4}. The partial line graphs of the LR-structures on the wreath graphs defined above are thus arc-transitive tetravalent graphs, which may also be obtained as 2-fold covers over the wreath graphs. These graphs have appeared in the literature before, for example in [6, 7]. 6.2 Toroidal LR Structures Consider the LR structure on the cube Q4 depicted in the left-hand side of Figure 3. This structure can be generalized to other maps on the torus: Definition 6.2. Let M be a map of type {4,4} on the torus. Then M arises as a quotient {4, 4}T of the tessellation {4,4} by a group T of translations. Color edges of the tessellation belonging to one parallel class green, the other edges red. Since translations preserve these colors, we can consider the edges of M colored in the same way. If the group T is normalized by horizontal and vertical reflections, we will say that T is reflective. In this case, then those reflections act on M as well, giving the structure its swappers. In any map of type {4,4} on the torus, the group of translations is transitive on vertices, and so if T is reflective, the resulting graph {4,4}T is an LR structure, and we will call it a toroidal LR structure. Note that if (A, C) is a toroidal LR structure, then P(A, C) is a bipartite edge-transitive skeleton of a map of type {4,4} on the torus. Remark: The possibilities for the group T of translations satisfying the requirement that T is normalized by all horizontal and vertical reflections are exactly these two: (I) T is generated by (b, 0) and (0, c) for some b and c both at least 3. The resulting graph has bc vertices and is of type {b, c}. We will call this structure {4,4}[b c]. (II) T is generated by (b, c) and (b, -c) for some b and c both at least 2. The resulting graph has 2bc vertices and is of type {2b, 2c}. We will call this structure {4,4}. These two types are shown in Figure 4; on the left is {4,4}[4,3], and on the right is {4, 4}<3,2>. Observe that every toroidal LR structure, as defined above, has alternating 4-cycles. Surprisingly, the converse holds as well. Lemma 6.3. An LR structure which has an alternating 4-cycle is toroidal and arises as in Definition 6.2. Proof. Suppose that a, b, c, d is an alternating 4-cycle in which {a, b} and {c, d} are the green edges, {b, c} and {d, a} the red. Then a green swapper at a fixes a and d but not b, and so sends abcd to a different alternating 4-cycle containing the red edge {d, a}. Thus every red edge belongs to at least two alternating 4-cycles, and similarly, so does every green. Now if any edge belongs to more than two alternating 4-cycles, it is not hard to show that the structure must be the toroidal structure for the map {4,4}2 2 (its underlying P. Potocnik and S. E. Wilson: Linking Rings Structures and tetravalent semisymmetric. 347 1 2 3 4 1 4 1 6 1 7 7 3 4 1 Figure 4: Two kinds of toroidal LR structures graph is K4 4). If not, then every edge belongs to exactly two alternating 4-cycles. Using these cycles as faces constructs a map on a surface. The map has quadrangular faces, and four meet at every vertex and so the Euler characteristic of the surface is 0. The surface is thus the torus or the Klein bottle. The results of the paper [21] show that no map on the Klein bottle can have swappers except for some whose skeleton is not a simple graph, and therefore the map and structure must be toroidal. □ This lemma, combined with [20, Theorem 5.1(5, b)], proves Theorem 5.2 6.3 Barrels In this section, we introduce our first families of suitable LR structures. The barrels Br(k, n; r) and the mutant barrels MBr(k, n; r) are, of all known LR structures, the simplest to describe, the easiest to visualize and the most common to occur. The structure Br(k, n; r) is defined for k even, k > 4, n > 5, and r2 = ±1 mod n but not r = ±1 mod n. It has vertices Zk x Zn, with (i, j) red-adjacent to (i ± 1, j) and green-adjacent to (i, j ± r®). We may and usually do assume that 0 < r < n. We should mention that the underlying graphs of the LR structures Br(k, n; r) have appeared in different contexts under different names before, most recently in [16], where the class of Y graphs include the graph Br(k, n; r) as Y(k, n, r, 0). Figure 5 shows part of such a graph having n = 5 and r = 2, with red edges shown thin, green edges shown bold. Br(k,n; r), has color-preserving symmetries (i,j) ^ (i,j + 1) and (i,j) ^ (i+1,rj), which show it to be vertex transitive. In the stabilizer of (0,0) are symmetries (i, j) ^ (—i, j) and (i,j) ^ (i,-j) which act as swappers there. Thus, Br(k, n; r) is an LR 340 Ars Math. Contemp. 7 (2014) 337-340 structure. A related graph is MBr(k, n; r), which is defined if both k and n are even, k > 2, n > 6, and r2 = ±1 mod n but not r = ±1 mod n. It has vertices Zk x Zn, with (i, j) red-adjacent to (i + 1, j) for 0 < i < k - 1, (k - 1, j) red-adjacent to (0, j + n/2) and (i, j) green-adjacent to (i, j ± r®) for all i. MBr(k,n; r) has symmetries that are swappers with definitions similar to those for Br(k, n; r). Thus, every member of these two families is an LR structure. For most parameters k, n and r, the LR structures Br(k, n; r) and MBr(k, n; r) are suitable. Each one has kn vertices and is of type {k, n}, {2k, n}, respectively. We are now ready to determine which barrels give rise to suitable LR-structures. Theorem 6.4. Each of the LR structures Br(k, n; r), MBr(k, n; r) has kn vertices and is of type {k,n}, {2k, n}, respectively. The LR structures Br(k, n; r) are suitable for all admissible parameters: k > 4 even, n > 5, and 2 < r < n, r2 = ±1 (mod n). The LR structure MBr(k, n; r) is suitable for all admissible parameters: k > 4 even, n > 6 even, 2 < r < n, r2 = ±1 (mod n), with the exceptions of MBr(k, 2k; k — 1) for k even, k > 4, which is self-dual; and MBr(2, n; n ± 1), which has alternating 4-cycles. Proof. The claim about the type of Br(k, n; r) and MBr(k, n; r) follows immediately from the definition of the two structures. It is also clear that no member of either family has an alternating 4-cycle, except in the case of MBr(2,n; n ± 1); here, an alternating 4-cycle is (0,0) - (0,1) - (1,1 + n) - (1,1 + n + r) = (1,0) - (0,0). Number the red cycles 1, 2,... in the order in which they appear around a fixed green cycle, and do the same for green cycles. In Br(k, n; r), every green cycle appears once around each red cycle and vice versa; in MBr(k, n; r), every green cycle appears twice around each red cycle and vice versa. Then in every red cycle, the green cycles appear in the same order, 1, 2,..., while around alternating green cycles, they appear in the orders 1,2,3 ... and 1,1 + r, 1 + 2r,.... In most cases, these are different orders, and so no symmetry can switch red with green. The only way that these can be the same order is if: (1) each number appears twice, so the graph is the underlying graph of a mutant barrel LR structure, (2) the cycles have the same length, so that n = 2k, P. Potocnik and S. E. Wilson: Linking Rings Structures and tetravalent semisymmetric. 349 (3) r + 1 = 2 mod k, so that r = k + 1 It remains to show that MBr(k, 2k; k - 1) is indeed self-dual. To do that, we observe that it is isomorphic to the following LR structure: vertices are elements of Zk x Zk x Z2; red edges are of the form: • (i, j, d) - (i + 1,j,d), 0 < i is antipodally attached, and the projection sends it onto {4,4}[b c] (if b and c are both at least 3). Each MBr(2k, 2n; r) is antipodally attached, and the projection sends it onto Br(k, n; r) (if k and n are both at least 3). If (A, C) is an antipodally attached LR structure of type {4,4}, then it is easy to see that A is a wreath graph with C the decomposition presented in Construction 6.1. Antipodally attached LR structures of type {4, q} for some q > 4 will be studied in a future paper, where we show that such a structure arises from a certain cycle decomposition, called a "cycle structure", in a smaller tetravalent dart-transitive graph, thus allowing a reduction to smaller graphs. The above discussion is summarized in the following theorem. Theorem 7.1. Let (A, C) be an antipodally attached LR structure of type {p, q}. If p = q = 4, then (A, C) is the type {4,4} LR structure on a wreath graph W(n, 2) described in Construction 6.1. If p, q > 6, then (A, C) arises as a 2-fold cover of a loosely attached LR-structure. Question 2: For which loosely attached LR structures does an antipodally attached cover exist? Question 3: How is such a cover constructed? Question 4: If (A, C) is suitable and antipodally attached, when is the factor structure suitable? P. Potocnik and S. E. Wilson: Linking Rings Structures and tetravalent semisymmetric. 351 8 The Big question In Section 5, we conjectured that a suitable LR structure is unique on its graph. We believe that a stronger conjecture holds: Conjecture 8.1. If (A, C) is an LR structure for which Aut+(A, C) is a proper subgroup of Aut(A), then it is self-dual. Certainly, if Aut+(A, C) is a proper subgroup of Aut(A, C), then (A, C) is self-dual. So we may suppose that Aut(A) contains some a which sends some red edges to red edges and some red edges to green edges. The conjecture would imply that (A, C) is self-dual. While that conjecture is still open, we can and will prove that such an LR structure must be of a self-dual type. Theorem 8.2. If (A, C) is an LR structure of type {p,q} and Aut(A) contains some a which does not preserve C, then p = q. Proof. We will actually prove that under the hypothesis, some symmetry must send some red cycle to a green cycle. Suppose not. Let m be maximal such that some a in Aut(A) sends some m consecutive edges of a red cycle to green edges. Then m must be less than both p and q. Suppose that u0, ui,..., um-1,um, um+1,... are consecutive vertices on some red cycle, v0, vi,..., vm-1,vm, vm+i,... are consecutive vertices on some green cycle, and that for i = 0,1, 2 ..., m, u^a = vj. Suppose that a, b are the green neighbors of um, that c, d are the red neighbors of vm. Finally, let ^ be a green swapper at um and t be a red swapper at vm, as shown in Figure 6. Figure 6: A color-mixing symmetry a Because m is maximal, we know that um+1a = vm+1. Assume, without loss of generality, that um+1a = c,aa = vm+1,ba = d. Consider a' = ara-1^a. 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