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Hamiltonian Cycles in Cubic Cayley Graphs: the 2, 4K,3 Case

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Hamiltonian Cycles in Cubic Cayley Graphs: the �2, 4K,3� Case J Algebr Comb (2009) 30: 447–475 DOI 10.1007/s10801-009-0172-5 Hamiltonian cycles in cubic Cayley graphs: the 2, 4k,3 case Henry H. Glover · Klavdija Kutnar · Dragan Marušicˇ Received: 22 May 2008 / Accepted: 16 February 2009 / Published online: 10 March 2009 © Springer Science+Business Media, LLC 2009 Abstract It was proved by Glover and Marušicˇ (J. Eur. Math. Soc. 9:775–787, 2007), that cubic Cayley graphs arising from groups G =a,x | a2 = xs = (ax)3 = 1,... having a (2,s,3)-presentation, that is, from groups generated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4, and have a Hamiltonian path when |G| is congruent to 0 modulo 4. In this article the existence of a Hamiltonian cycle is proved when apart from |G| also s is congruent to 0 modulo 4, thus leaving |G| congruent to 0 modulo 4 with s either odd or congruent to 2 modulo 4 as the only remaining cases to be dealt with in order to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs. Keywords Hamiltonian cycle · Cayley graph · Cubic arc-transitive graph · Consistent cycle · Cyclic edge connectivity 1 Introductory remarks In 1969, Lovász [27] asked if every finite, connected vertex-transitive graph has a Hamiltonian path, that is, a simple path going through all vertices of the graph. With H.H. Glover Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, USA K. Kutnar University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia D. Marušic(ˇ ) University of Ljubljana, IMFM, Jadranska 19, 1000 Ljubljana, Slovenia e-mail: [email protected] D. Marušicˇ University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia 448 J Algebr Comb (2009) 30: 447–475 the exception of K2, only four connected vertex-transitive graphs that do not have a Hamiltonian cycle are known to exist. These four graphs are the Petersen graph, the Coxeter graph and the two graphs obtained from them by replacing each vertex by a triangle. The fact that none of these four graphs is a Cayley graph has led to a folklore conjecture that every Cayley graph with order greater than 2 has a Hamiltonian cycle. Despite the fact that these questions have been challenging mathematicians for almost forty years, only partial results have been obtained thus far. For instance, it is known that connected vertex-transitive graphs of orders kp, where k ≤ 5, pj , where j ≤ 4, and 2p2 contain a Hamiltonian path. Furthermore, for all of these families, except for the graphs of order 5p, it is also known that they contain a Hamiltonian cycle (except for the above mentioned Petersen and Coxeter graph), see [1, 6, 25, 29– 33, 38]. For the subclass of Cayley graphs a number of partial results are known (see for example [2, 12, 18, 19, 24, 28, 39, 40]). Also, it is known that every connected vertex-transitive graph, other than the Petersen graph, whose automorphism group contains a transitive subgroup with a cyclic commutator subgroup of prime-power order, has a Hamiltonian cycle. The result was proved in [14] and it uses results from a series of papers dealing with the same group-theoretic restrictions in the context of Cayley graphs [13, 28, 39]. It was proved in [18] that cubic Cayley graphs arising from groups having a (2,s,3)-presentation, that is, for groups G =a,x | a2 = xs = (ax)3 = 1,... gen- erated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4 and have a Hamiltonian path when |G| is congruent to 0 modulo 4. In this paper we proved the existence of a Hamiltonian cycle in a class of cubic Cayley graphs arising from groups having a (2,s,3)-presentation when apart from |G| also s is congruent to 0 modulo 4, thus leaving |G| congruent to 0 modulo 4 with s either odd or congruent to 2 modulo 4 as the only remaining cases to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs. The following is therefore the main result of this paper. Theorem 1.1 Let s ≡ 0 (mod 4) be a positive integer and let G =a,x | a2 = xs = (ax)3 = 1,... be a group with a (2,s,3)-presentation. Then the Cayley graph Cay(G,{a,x,x−1}) has a Hamiltonian cycle. For the motivation for studying this family of graphs see [18]. This article is organized as follows. In Section 2 notation and terminology is in- troduced and certain results, essential to the strategy of the proof of Theorem 1.1,are gathered. In Section 3 examples illustrating the method of the proof of Theorem 1.1 are given. The proof of Theorem 1.1 is carried out in Section 6. Our strategy is based on an embedding of X = Cay(G, S) in the closed orientable surface with s-gonal and hexagonal faces, in which we then look for a Hamiltonian tree of faces, that is, a tree of faces whose boundary is a Hamiltonian cycle in X. This Hamiltonian tree of faces is obtained in the following way. First, we associate to X asocalledhexagon graph Hex(X), a cubic graph whose vertex set consists of all hexagons arising from the relation (ax)3 with two such hexagons being adjacent if they share an edge in X (see Section 3 for more details). Second, we modify Hex(X) somewhat by deleting J Algebr Comb (2009) 30: 447–475 449 some of its vertices and edges to end up with a smaller cubic graph Mod(X) whose order is congruent to 2 modulo 4. Third, we find a subset of the vertex set of Mod(X) inducing a tree, whose complement is an independent set of vertices. This is possible by showing that Mod(X) is cyclically 4-edge connected (carried out in Sections 4 and 5 following a tedious analysis of various possibilities that can occur for Hex(X)) and then using the result of Payan and Sakarovitch given in Proposition 2.1. Finally, the above tree in Mod(X) is then used to produce a Hamiltonian tree of faces in the embedding of X, and hence a Hamiltonian cycle in X (two examples illustrating this method are given in Section 3). 2 Preliminaries 2.1 Notation Given a graph X we let V(X), E(X), A(X) and Aut X be the vertex set, the edge set, the arc set and the automorphism group of X, respectively. For adjacent vertices u and v in X, we write u ∼ v and denote the corresponding edge by uv.Ifu ∈ V(X) then N(u) denotes the set of neighbors of u and Ni(u) denotes the set of vertices at distance i>1 from u. A graph X is said to be cubic if |N(u)|=3 for any vertex u ∈ V(X). A sequence (u0,u1,u2,...,uk) of distinct vertices in a graph is called a k-arc if ui is adjacent to ui+1 for every i ∈{0, 1,...,k− 1}.ForS ⊆ V(X) we let X[S] denote the induced subgraph of X on S.Byann-cycle we shall always mean a cycle with n vertices. We will use the symbol Zn, both for the cyclic group of order n Z∗ and the ring of integers modulo n. In the latter case, n will denote the multiplicative group of units of Zn. A subgroup G ≤ Aut X is said to be vertex-transitive, edge-transitive and arc- transitive provided it acts transitively on the sets of vertices, edges and arcs of X,re- spectively. A graph is said to be vertex-transitive, edge-transitive, and arc-transitive if its automorphism group is vertex-transitive, edge-transitive and arc-transitive, re- spectively. An arc-transitive graph is also called symmetric. A subgroup G ≤ Aut X is said to be k-regular if it acts transitively on the set of k-arcs and the stabilizer of a k-arc in G is trivial. Given a group G and a generating set S of G such that S = S−1 and 1 ∈/ S,the Cayley graph Cay(G, S) of G relative to S has vertex set G and edge set {g ∼ gs | g ∈ G, s ∈ S}. It is easy to see that the group G acts regularly on the vertex set G by left multiplication, and hence G may be viewed as a regular subgroup of the automorphism group of the Cayley graph X = Cay(G, S). 2.2 Cyclic stability and cyclic connectivity Following [36] we say that, given a graph (or more generally a loopless multi- graph) X, a subset S of V(X) is cyclically stable if the induced subgraph X[S] is acyclic, that is, a forest. The size |S| of a maximum cyclically stable subset S of V(X)is called the cyclic stability number of X. Given a connected graph X, a subset F ⊆ E(X) of edges of X is said to be cycle- separating if X −F is disconnected and at least two of its components contain cycles. 450 J Algebr Comb (2009) 30: 447–475 We say that X is cyclically k-edge-connected if no set of fewer than k edges is cycle- separating in X.
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