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Homogeneous graphs and stability

Published online by Cambridge University Press:  09 April 2009

A. Gardiner
Affiliation:
Pure Mathematics, University of Birmingham, Birmingham B15 2TT, England.
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Abstract

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Let Γ be a graph with finite vertex set V. Γ is homogeneous if whenever U1, U2V 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 UV, then every automorphism of (U) extends to an automorphism of Γ. We prove that every locally homogeneous graph is homogeneous.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1976

References

Gardiner, A. (to appear), ‘Homogeneous graphs’, J. Combinatorial Theory.Google Scholar
Sheehan, J. (1974), ‘Smoothly embeddable subgraphs’, J. London Math. Soc. (2), 9, 212218.CrossRefGoogle Scholar
Wilson, R. J. (1972), Introduction to graph theory (Oliver and Boyd, Edinburgh).Google Scholar
Wielandt, H. (1964), Finite permutation groups (Academic Press, New York).Google Scholar
Yap, H. P. (1974), ‘Some remarks on stable graphs’, Bull. Austral. Math. Soc. 10, 351357.CrossRefGoogle Scholar