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Some remarks on universal graphs

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Abstract

LetΓ be a class of countable graphs, and let ℱ(Γ) denote the class of all countable graphs that do not contain any subgraph isomorphic to a member ofΓ. Furthermore, letTΓ andHΓ denote the class of all subdivisions of graphs inΓ and the class of all graphs contracting to a member ofΓ, respectively. As the main result of this paper it is decided which of the classes ℱ(TK n) and ℱ(HK n),n≦ℵ0, contain a universal element. In fact, for ℱ(TK 4)=ℱ(HK 4) a strongly universal graph is constructed, whereas for 5≦n≦ℵ0 the classes ℱ(TK n) and ℱ(HK n) have no universal elements.

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Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, England

    Reinhard Diestel

  2. Mathematisches Seminar, Universität Hamburg, Bundesstraße 55, 2000, Hamburg 13, Germany

    Rudolf Halin

  3. Fachbereich Informatik, Universität Hamburg, Rothenbaumchaussee 67/69, Germany

    Walter Vogler

Authors
  1. Reinhard Diestel
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  2. Rudolf Halin
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  3. Walter Vogler
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Dedicated to Klaus Wagner on his 75th birthday

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Diestel, R., Halin, R. & Vogler, W. Some remarks on universal graphs. Combinatorica 5, 283–293 (1985). https://doi.org/10.1007/BF02579242

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  • DOI: https://doi.org/10.1007/BF02579242

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