  The main result of this paper is that, if $\Gamma$ is a finite connected
$4$-valent arc-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 vertices. As a corollary, we get a similar
result for $3$-valent vertex-transitive graphs. Based on these results we
propose a conjecture on the number of fixed points of non-identity
automorphisms of vertex-transitive graphs of bounded valency.
