On $k$-connected-homogeneous graphs

TL;DR

This paper investigates the structure of $k$-connected-homogeneous graphs, classifies certain locally finite cases, and explores conditions under which these graphs are fully homogeneous, revealing new classifications and properties.

Contribution

It classifies locally finite, locally connected 4-CH graphs and explores conditions for full homogeneity in finite 7-CH graphs, advancing understanding of $k$-connected-homogeneity.

Findings

Classified locally finite, locally connected 4-CH graphs.

Classified locally finite, locally disconnected 4-CH graphs with specific cycle properties.

Most finite 7-CH graphs are fully homogeneous, with some exceptions.

Abstract

A graph $\Gamma$ is $k$-connected-homogeneous ($k$-CH) if $k$ is a positive integer and any isomorphism between connected induced subgraphs of order at most $k$ extends to an automorphism of $\Gamma$, and connected-homogeneous (CH) if this property holds for all $k$. Locally finite, locally connected graphs often fail to be 4-CH because of a combinatorial obstruction called the unique $x$ property; we prove that this property holds for locally strongly regular graphs under various purely combinatorial assumptions. We then classify the locally finite, locally connected 4-CH graphs. We also classify the locally finite, locally disconnected 4-CH graphs containing 3-cycles and induced 4-cycles, and prove that, with the possible exception of locally disconnected graphs containing 3-cycles but no induced 4-cycles, every finite 7-CH graph is CH.