On 2-arc-transitivity of Cayley graphs

  • Authors:

  • Dragan Marušič

  • Affiliations:

  • IMFM, Oddelek za Matematiko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenia

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2003

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Abstract

The classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra. Combin. 5 (1996) 83- 86) by Alspach. Conder, Xu and the author, motivates the main theme of this article: the study of 2-arc-transitive Cayley graphs of dihedral groups. First, a previously unknown infinite family of such graphs, arising as covers of certain complete graphs, is presented, leading to an interesting property of Singer cycles in the group PGL(2, q), q an odd prime power, among others. Second, a structural reduction theorem for 2-arc-transitive Cayley graphs of dihedral groups is proved, putting us--modulo a possible existence of such graphs among regular cyclic covers over a small family of certain bipartite graphs--a step away from a complete classification of such graphs. As a byproduct, a partial description of 2- arc-transitive Cayley graphs of abelian groups with at most three involutions is also obtained.