## Abstract

A graph Γ 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 Γ, 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.

Original language | English |
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Article number | 105234 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 173 |

DOIs | |

Publication status | Published - 1 Jul 2020 |