Ramsey Numbers for Line Graphs and Perfect Graphs

  • Rémy Belmonte
  • Pinar Heggernes
  • Pim van ’t Hof
  • Reza Saei
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

For any graph class \({\cal G}\) and any two positive integers i and j, the Ramsey number \(R_{\cal G}(i,j)\) is the smallest integer such that every graph in \({\cal G}\) on at least \(R_{\cal G}(i,j)\) vertices has a clique of size i or an independent set of size j. For the class of all graphs Ramsey numbers are notoriously hard to determine, and the exact values are known only for very small integers i and j. For planar graphs all Ramsey numbers can be determined by an exact formula, whereas for claw-free graphs there exist Ramsey numbers that are as difficult to determine as for arbitrary graphs. No further graph classes seem to have been studied for this purpose. Here, we give exact formulas for determining all Ramsey numbers for various classes of graphs. Our main result is an exact formula for all Ramsey numbers for line graphs, which form a large subclass of claw-free graphs and are not perfect. We obtain this by proving a general result of independent interest: an upper bound on the number of edges any graph can have if it has bounded degree and bounded matching size. As complementary results, we determine all Ramsey numbers for perfect graphs and for several subclasses of perfect graphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. SIAM (1999)Google Scholar
  2. 2.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chudnovsky, M., Seymour, P.: The structure of claw-free graphs. In: Surveys in Combinatorics 2005. London Math. Soc. Lecture Note Ser., p. 327 (2005)Google Scholar
  4. 4.
    Cockayne, E.J., Lorimer, P.J.: On Ramsey graph numbers for stars and stripes. Canad. Math. Bull. 18, 31–34 (1975)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory, Electronic edition. Springer (2005)Google Scholar
  6. 6.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Disc. Math. 57 (2004)Google Scholar
  7. 7.
    Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory, 2nd edn. Wiley (1990)Google Scholar
  8. 8.
    Harary, F.: Graph Theory. Addison-Wesley (1969)Google Scholar
  9. 9.
    Matthews, M.M.: Longest paths and cycles in K 1,3-free graphs. Journal of Graph Theory 9, 269–277 (1985)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Radziszowski, S.P.: Small Ramsey numbers. Electronic Journal of Combinatorics, Dynamic Surveys (2011)Google Scholar
  11. 11.
    Ramsey, F.P.: On a problem of formal logic. Proc. London Math. Soc. Series 2, vol. 30, pp. 264–286 (1930)Google Scholar
  12. 12.
    Spencer, J.H.: Ten Lectures on the Probabilistic Method. SIAM (1994)Google Scholar
  13. 13.
    Steinberg, R., Tovey, C.A.: Planar Ramsey numbers. J. Combinatorial Theory Series B 59, 288–296 (1993)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Vizing, V.G.: On an estimate of the chromatic class of a p-graph. Diskret. Analiz. 3, 25–30 (1964) (in Russian)MathSciNetGoogle Scholar
  15. 15.
    Walker, K.: The analog of Ramsey numbers for planar graphs. Bull. London Math. Soc. 1, 187–190 (1969)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Rémy Belmonte
    • 1
  • Pinar Heggernes
    • 1
  • Pim van ’t Hof
    • 1
  • Reza Saei
    • 1
  1. 1.Department of InformaticsUniversity of BergenNorway

Personalised recommendations