Some remarks on universal graphs


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, let and 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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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).

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AMS subject classification (1980)

  • 05 C 75