On intransitive graph-restrictive permutation groups

TL;DR

This paper characterizes intransitive permutation groups that are graph-restrictive, showing they are exactly the semiregular groups, thus advancing understanding of symmetry constraints in vertex-transitive graphs.

Contribution

It proves that intransitive groups are graph-restrictive if and only if they are semiregular, providing a complete characterization of such groups.

Findings

Intransitive groups are graph-restrictive iff they are semiregular.

Provides a necessary and sufficient condition for intransitive groups to be graph-restrictive.

Enhances understanding of symmetry restrictions in vertex-transitive graphs.

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 $|G_v|\leq c(L)$ holds. We show that an intransitive group is graph-restrictive if and only if it is semiregular.