  Let $\Gamma$ be a connected $G$-arc-transitive graph, let $uv$ be an arc of
$\Gamma$ and let $L$ be the permutation group induced by the action of the
vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. We study the problem
of bounding $|G_{uv}|$ in terms of $L$ and the order of $\Gamma$.
