# Locally finite homogeneous graphs

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DOI: 10.1007/s00493-010-2472-8

- Cite this article as:
- Hedman, S. & Pong, W.Y. Combinatorica (2010) 30: 419. doi:10.1007/s00493-010-2472-8

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## Abstract

A connected graph *G* is said to be *z*-homogeneous if any isomorphism between finite connected induced subgraphs of *G* extends to an automorphism of *G*. Finite *z*-homogeneous graphs were classified in [17]. We show that *z*-homogeneity is equivalent to finite-transitivity on the class of infinite locally finite graphs. Moreover, we classify the graphs satisfying these properties. Our study of bipartite *z*-homogeneous graphs leads to a new characterization for hypercubes.