The structure of hereditary properties and 2-coloured multigraphs

Abstract

In “The structure of hereditary properties and colourings of random graphs” [Combinatorica 20(2) (2000), 173–202], Bollobás and the second author studied the probability of a hereditary property P in the probability space G(n,p). They found simple properties that closely approximate P in this space, and using these simple properties they determined the P-chromatic number of random graphs.

In this note we point out that the analysis of hereditary properties in G(n,p) can be made more exact by means of the extremal properties of 2-coloured multigraphs, and we illustrate some cases in which the probability of P can thereby be calculated. At the same time we correct a mistake in the cited paper.

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References

  1. [1]

    V. E. Alekseev: On the entropy values of hereditary classes of graphs, Discrete Math. Appl. 3 (1993), 191–199.

    MathSciNet  Article  Google Scholar 

  2. [2]

    N. Alon and U. Stav: What is the furthest graph from a hereditary property?, Random Structures and Algorithms 33 (2008), 87–104.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    M. Axenovich, A. Kézdy and R. Martin: On the editing distance of graphs, J. Graph Theory 58 (2008), 123–138.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    J. Balogh and R. Martin: Edit distance and its computation, Electronic Journal of Combinatorics 15 (2008), #R20.

    Google Scholar 

  5. [5]

    B. Bollobás and A. Thomason: Projections of bodies and hereditary properties of hypergraphs, J. London Math. Soc. 27 (1995), 417–424.

    MATH  Article  Google Scholar 

  6. [6]

    B. Bollobás and A. Thomason: Hereditary and monotone properties of graphs, in: The Mathematics of Paul Erdős II (R. L. Graham and J. Nešetřil, eds.), Algorithms and Combinatorics 14, Springer-Verlag, (1997), 70–78.

  7. [7]

    B. Bollobás and A. Thomason: The structure of hereditary properties and colourings of random graphs, Combinatorica 20(2) (2000), 173–202.

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    W. G. Brown, P. Erdős and M. Simonovits: Extremal problems for directed graphs, J. Combinatorial Theory (Ser. B) 15 (1973), 77–93.

    MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    W. G. Brown, P. Erdős and M. Simonovits: Inverse extremal digraph problems, in: Finite and Infinite Sets, Eger (Hungary), 1981, Colloq. Math. Soc. János Bolyai 37, Akad. Kiadó, Budapest (1985), 119–156.

    Google Scholar 

  10. [10]

    W. G. Brown, P. Erdős and M. Simonovits: Algorithmic solution of extremal digraph problems, Trans. Amer. Math. Soc. 292 (1985), 421–449.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    E. Marchant and A. Thomason: Extremal graphs and multigraphs with two weighted colours, in: Fete of Combinatorics and Computer Science, Bolyai Soc. Math. Stud. 20, (2010), 239–286.

    Article  Google Scholar 

  12. [12]

    H. J. Prömel and A. Steger: The asymptotic structure of H-free graphs, in: Graph Structure Theory (N. Robertson and P. Seymour, eds), Contemporary Mathematics 147, Amer. Math. Soc., Providence, 1993, pp. 167–178.

    Google Scholar 

  13. [13]

    D. C. Richer: Ph.D. thesis, University of Cambridge (2000).

  14. [14]

    U. Stav: personal communication.

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Correspondence to Edward Marchant.

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Research funded by Trinity College, University of Cambridge.

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Marchant, E., Thomason, A. The structure of hereditary properties and 2-coloured multigraphs. Combinatorica 31, 85 (2011). https://doi.org/10.1007/s00493-011-2630-7

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Mathematics Subject Classification (2000)

  • 05C35
  • 05C80