On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency

Abstract

We prove that, if Γ is a finite connected 3-valent vertex-transitive, or 4-valent vertex-and edge-transitive graph, then either Γ is part of a well-understood family of graphs, or every non-identity automorphism of Γ fixes at most 1/3 of the edges. This answers a question proposed by Primož Potočnik and the third author.