Homogeneous graphs and stability

Abstract

AbstractLet Γ be a graph with finite vertex set V. Γ is homogeneous if whenever U1, U2 ⊆ V are such that the vertex subgraphs (U1), (U2) are isomorphic, then every isomorphism from (U1) to (U2) extends to an automorphism of Γ; homogeneous graphs were studied by Sheehan (1974) and were classified by the author. Γ is locally homogeneous if whenever U ⊆ V, then every automorphism of (U) extends to an automorphism of Γ. We prove that every locally homogeneous graph is homogeneous.