Some remarks on universal graphs

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, 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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    B. Bollobás,Extremal Graph Theory, Academic Press, London 1978.

    MATH  Google Scholar 

  2. [2]

    R. Diestel, On Universal Graphs With Forbidden Topological Subgraphs, to appear inEurop. J. Combinatorics.

  3. [3]

    R. Diestel, On the Problem of Finding Small Subdivision and Homomorphism Bases for Classes of Countable Graphs, to appear inDiscrete Math.

  4. [4]

    R. Diestel, Simplicial Decompositions of Graphs — Some Uniqueness Results, submitted.

  5. [5]

    R. Halin,Graphentheorie II, Wissenschaftliche Buchgesellschaft, Darmstadt 1981.

  6. [6]

    R. Halin, Zur Klassifikation der endlichen Graphen nach H. Hadwiger und K. Wagner.Math. Ann.172 (1967), 46–78.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    R. Halin andH. A. Jung, Über Minimalstrukturen von Graphen, insbesondere vonn-zusammenhängenden Graphen,Math. Ann.152 (1963), 75–94.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    P. Komjáth andJ. Pach, Universal Graphs Without Large Bipartite Subgraphs,Mathematika,31 (1984), 282–290.

    MATH  MathSciNet  Article  Google Scholar 

  9. [9]

    J. Pach, A Problem of Ulam on Planar Graphs,Europ. J. Combinatorics2 (1982), 357–361.

    MathSciNet  Google Scholar 

  10. [10]

    R. Rado, Universal Graphs and Universal Functions,Acta Arith.9 (1964), 331–340.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Dedicated to Klaus Wagner on his 75th birthday

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

AMS subject classification (1980)

  • 05 C 75