18 research outputs found
Resolution of a conjecture about linking ring structures
An LR-structure is a tetravalent vertex-transitive graph together with a
special type of a decomposition of its edge-set into cycles. LR-structures were
introduced in a paper by P. Poto\v{c}nik and S. Wilson, titled `Linking rings
structures and tetravalent semisymmetric graphs', in Ars Math. Contemp. 7
(2014), as a tool to study tetravalent semisymmetric graphs of girth 4. In this
paper, we use the methods of group amalgams to resolve some problems left open
in the above-mentioned paper
On 2-arc-transitivity of Cayley graphs
AbstractThe classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra. Combin. 5 (1996) 83ā86) 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 usāmodulo a possible existence of such graphs among regular cyclic covers over a small family of certain bipartite graphsāa 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