On the order of arc-stabilisers in arc-transitive graphs, II

Let $\Gamma$ be a $G$-vertex-transitive graph and let $(u,v)$ be an arc of $\Gamma$. It is known that if the local action $G_v^{\Gamma(v)}$ (the permutation group induced by $G_v$ on $\Gamma(v)$) is permutation isomorphic to the dihedral group of degree $4$, then either $|G_{uv}|$ is ``small" with respect to the order of $\Gamma$ or $\Gamma$ is one of a family of well understood graphs. In this paper, we generalise this result to a wider class of local actions. 10.1017/S000497271200047

Semiregular automorphisms of cubic vertex-transitive graphs and the abelian normal quotient method

Sherpa Romeo green journal. Open accessWe characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.Ye