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.
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