On intransitive graph-restrictive permutation groups

Abstract

Let $\Gamma$ be a finite connected $G$-vertex-transitive graph and let $v$ be a vertex of $\Gamma$. If the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma (v)$ is permutation isomorphic to $L$, then $(\Gamma ,G)$ is said to be locally-$L$. A permutation group $L$ is graph-restrictive if there exists a constant $c(L)$ such that, for every locally-$L$ pair $(\Gamma ,G)$ and a vertex $v$ of $\Gamma$, the inequality $|G_v|\le c(L)$ holds. We show that an intransitive group is graph-restrictive if and only if it is semiregular.

6 pages, 3 figures