On 3-arc-transitive covers of the dodecahedron graph

Abstract

In this paper, the following problem is considered: does there exist a t-arc-transitive regular covering graph of an s-arc-transitive graph for positive integers t greater than s? In order to answer this question, we classify all arc-transitive cyclic regular covers of the dodecahedron graph. Two infinite families of 3-arc-transitive abelian covering graphs are given, which give more specific examples that for an s-arc-transitive graph there exist (s+1)-arc-transitive covering graphs.