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

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