On intransitive graph-restrictive permutation groups

Abstract

Let $Γ$ be a finite connected $G$-vertex-transitive graph and let $v$ be a vertex of $Γ$. If the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$ is permutation isomorphic to $L$, then $(Γ,G)$ 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 $(Γ,G)$ and a vertex $v$ of $Γ$, the inequality $|G_v|\leq c(L)$ holds. We show that an intransitive group is graph-restrictive if and only if it is semiregular.

6 pages, 3 figures