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

TL;DR

This paper establishes bounds on the number of edges fixed by automorphisms in small valency vertex-transitive graphs, showing they are either well-characterized or have automorphisms fixing at most one-third of edges.

Contribution

It proves a new bound on fixed edges of automorphisms in 3- and 4-valent vertex-transitive graphs, addressing a question by Potočnik and the third author.

Findings

Automorphisms fix at most 1/3 of edges in certain graphs.

Identifies well-understood families of graphs in this context.

Provides a partial classification related to automorphism fixed edges.

Abstract

We prove that, if $\Gamma$ is a finite connected $3$-valent vertex-transitive, or $4$-valent vertex- and edge-transitive graph, then either $\Gamma$ is part of a well-understood family of graphs, or every non-identity automorphism of $\Gamma$ fixes at most $1/3$ of the edges. This answers a question proposed by Primo\v{z} Poto\v{c}nik and the third author.