Finite $3$-connected homogeneous graphs

TL;DR

This paper characterizes finite (G,3)-connected homogeneous graphs, showing they are either 2-transitive or of rank 3 with girth 3, and explores their structure and examples.

Contribution

It develops a new method for characterizing (G,3)-connected homogeneous graphs and classifies their quasiprimitive automorphism group types.

Findings

Either G_v^{Γ(v)} is 2-transitive or of rank 3 with girth 3.

The class of such graphs is closed under normal quotients.

New examples for certain quasiprimitive types are constructed.

Abstract

A finite graph $\G$ is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph $\G=(V, E)$, either $G_v^{\G(v)}$ is $2$--transitive or $G_v^{\G(v)}$ is of rank $3$ and $\G$ has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types.