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.

Author Keywords: Permutation group; Imprimitive group; Abelian group; Dihedral group; Cayley graph; 2-Arc-transitive graph

Article Outline

1. Introductory remarks

2. Laying out the strategy

3. Notation, terminology, examples

4. Graph coverings

5. The graphs K_q+1⁴ and Singer cycles

6. Abelian groups

7. Blocks and symbols of dihedrants

8. Proving Theorem 2.1

References

¹ Supported in part by “Ministrstvo za znanost in tehnologijo Slovenije,” Proj. No. J1-4965-01.