Symmetries in graphs via simplicial automorphisms

Abstract

An automorphism ρ of a graph X is said to be semiregular provided all of its cycles in its cycle decomposition are of the same length, and is said to be simplicial if it is semiregular and the quotient multigraph Xρ of X with respect to ρ is a simple graph, and thus of the same valency as X. It is shown that, with the exception of the complete graph K4, the Petersen graph, the Coxeter graph and the so called H-graph (alternatively denoted as S(17), the smallest graph in the family of the so called Sextet graphs S(p), p≡±1(mod16)), every cubic arc-transitive graph with a primitive automorphism group admits a simplicial automorphism.