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JCT

2010

2010

Abstract. A graph is connected-homogeneous if any isomorphism between ﬁnite connected induced subgraphs extends to an automorphism of the graph. In this paper we classify the countably inﬁnite connectedhomogeneous graphs. We prove that if Γ is connected countably inﬁnite and connected-homogeneous then Γ is isomorphic to one of: Lachlan and Woodrow’s ultrahomogeneous graphs; the generic bipartite graph; the bipartite ‘complement of a complete matching’; the line graph of the complete bipartite graph Kℵ0,ℵ0 ; or one of the ‘treelike’ distancetransitive graphs Xκ1,κ2 where κ1, κ2 ∈ N∪{ℵ0}. It then follows that an arbitrary countably inﬁnite connected-homogeneous graph is a disjoint union of a ﬁnite or countable number of disjoint copies of one of these graphs, or to the disjoint union of countably many copies of a ﬁnite connected-homogeneous graph. The latter were classiﬁed by Gardiner (1976). We also classify the countably inﬁnite connected-homog...

Related Content

Added |
28 Jan 2011 |

Updated |
28 Jan 2011 |

Type |
Journal |

Year |
2010 |

Where |
JCT |

Authors |
Robert Gray, D. Macpherson |

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