Extending partial isomorphisms of graphs


Let X be a finite graph. Then there exists a finite graph Z containing X as an induced subgraphs, such that every isomorphism between induced subgraphs of X extends to an automorphism of Z.

The graphZ may be required to be edge-transitive. The result implies that for anyn, there exists a notion of a “genericn-tuple of automorphism” of the Rado graphR (the random countable graph): for almost all automorphism σ1,..., σ n and τ1 ofR (in the sense of Baire category), (R1,...,σ n ), ≅ (R1,...,τ n ). The problem arose in a recent paper of Hodges, Hodgkinson, Lascar and Shelah, where the theorem is used to prove the small index property forR.

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  1. [1]

    L. Babai, andV. T. Sós: Sidon sets in groups and induced subgraphs of Cayley graphs,Europ. J. Comb. 6 (1985), 101–114.

    Google Scholar 

  2. [2]

    P. Cameron:Oligomorphic Permutation Groups, LMS 152, Cambridge University press, 1990.

  3. [3]

    P. Erdős, andA. Rényi: Asymmetric graphs,Acta Math. Acad. Sci. Hung. 14 (1963), 295–315.

    Google Scholar 

  4. [4]

    R. Fraïssé: Sur l'extension aux relations de quelques proprietes des ordres,Ann. Sci. École Norm. Sup. 71 (1954), 361–388.

    Google Scholar 

  5. [5]

    W. Hodges, I. Hodkinson, D. Lascar, andS. Shelah: The small index property for omega-stable omega-categorical structures and for the random graph, preprint.

  6. [6]

    D. Lascar: The group of automorphisms of a relational saturated structure, to appear in Proceedings of 1991 Banff conference on finite combinatorics.

  7. [7]

    J. Truss: Generic Automorphisms of Homogeneous Structures, to appear in LMS Proceedings.

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Work supported by a Sloan Fellowship and by NSF grant DMS-8903378.

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Hrushovski, E. Extending partial isomorphisms of graphs. Combinatorica 12, 411–416 (1992). https://doi.org/10.1007/BF01305233

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AMS subject classification code (1991)

  • 05 C 25