Quasi-semiregular automorphisms of cubic and tetravalent arc-transitive graphs

A non-trivial automorphism g of a graph Gamma is called semiregular if the only power g(i) fixing a vertex is the identity mapping, and it is called quasi-semiregular if it fixes one vertex and the only power g(i) fixing another vertex is the identity mapping. In this paper, we prove that K-4, the Petersen graph and the Coxeter graph are the only connected cubic arc-transitive graphs admitting a quasi-semiregular automorphism, and K-5 is the only connected tetravalent 2-arc-transitive graph admitting a quasi-semiregular automorphism. It will also be shown that every connected tetravalent G-arc-transitive graph, where G is a solvable group containing a quasi-semiregular automorphism, is a normal Cayley graph of an abelian group of odd order. (C) 2019 Elsevier Inc. All rights reserved.