Commutator - Automorphism groups with cyclic commutator...

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Unformatted text preview: Automorphism groups with cyclic commutator subgroup and Hamilton cycles Edward Dobson Heather Gavlas Department of Mathematics Department of Mathematics and Statistics Louisiana State University Grand Valley State University Baton Rouge, LA 70803 Allendale, MI 49401 U.S.A. U.S.A. Joy Morris Dave Witte Department of Mathematics and Statistics Department of Mathematics Simon Fraser University Oklahoma State University Burnaby, BC V5A 1S6 Stillwater, OK 74078 Canada U.S.A. Abstract It has been shown that there is a Hamilton cycle in every connected Cayley graph on any group G whose commutator subgroup is cyclic of prime-power order. This note considers connected, vertex-transitive graphs X of order at least 3, such that the automorphism group of X contains a vertex-transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs, only the Petersen graph is not hamiltonian. 1. Introduction Considerable attention has been devoted to the problem of determining whether or not a connected, vertex-transitive graph X has a Hamilton cycle [A1], [WG]. The vertex-transitivity implies that some group G of automorphisms of X acts transitively on V ( X ). If G can be chosen to be abelian, it is easy to see that X has a Hamilton cycle, so it is natural to try to prove the same conclusion when G is “almost abelian.” Thus, recalling that the commutator subgroup of G is the subgroup G = h x- 1 y- 1 xy : x,y ∈ G i , and that G is abelian if and only if the commutator subgroup of G is trivial, it is natural to consider the case where the commutator subgroup of G is “small” in some sense. In this vein, K. Keating and D. Witte [KW] used a method of D. Maruˇ siˇ c [M] to show that there is a Hamilton cycle in every Cayley graph on any group whose commutator subgroup is cyclic of prime-power order. This note utilizes techniques of B. Alspach, E. Durnberger, and T. Parsons [AP, ADP, A2] to extend this result to vertex-transitive graphs. Theorem 1.1. Let X be a connected vertex-transitive graph of order at least 3 . If there is a vertex-transitive group G of automorphisms of X , such that the commutator subgroup of G is cyclic of prime-power order, then X is the Petersen graph or X is hamiltonian. Because K 2 and the Petersen graph have Hamilton paths, the following corollary is immediate. Corollary 1.2. Let X be a connected vertex-transitive graph. If there is a vertex-transitive group G of automorphisms of X , such that the commutator subgroup of G is cyclic of prime-power order, then X has a Hamilton path. Acknowledgment. Much of this research was carried out at the Centre de Recherches Math´ ematiques of the Universit´ e de Montr´ eal. The authors would like to thank the organizers and participants of the Workshop 1 on Graph Symmetry, and the staff of the CRM, for the stimulating environment they provided. Witte was partially supported by a grant from the National Science Foundation....
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This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).

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Commutator - Automorphism groups with cyclic commutator...

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