Graphs and Combinatorics

, Volume 29, Issue 5, pp 1183–1191 | Cite as

Large Chromatic Number and Ramsey Graphs

Original Paper

Abstract

Let Q (n, χ) denote the minimum clique size an n-vertex graph can have if its chromatic number is χ. Using Ramsey graphs we give an exact, albeit implicit, formula for the case χ ≥ (n + 3)/2.

Keywords

Clique number Chromatic number Ramsey graphs 

Mathematics Subject Classification

05C69 05C35 05D10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ajtai M., Komlós J., Szemerédi E.: A note on Ramsey numbers. J. Comb. Theory Ser. A 29(3), 354–360 (1980)MATHCrossRefGoogle Scholar
  2. 2.
    Biró, C.: Large cliques in graphs with high chromatic number (2011). ArXiv:1107.2630Google Scholar
  3. 3.
    Bogart K.P., Trotter W.T.: Maximal dimensional partially ordered sets. II. Characterization of 2n-element posets with dimension n. Discret. Math. 5, 33–43 (1973)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Gyárfás, A., Sebő, A., Trotignon, N.: The chromatic gap and its extremes (2011). ArXiv:1108.3444Google Scholar
  5. 5.
    Hiraguchi T.: On the dimension of orders. Sci. Rep. Kanazawa Univ. 4(1), 1–20 (1955)MathSciNetGoogle Scholar
  6. 6.
  7. 7.
    Kim J.H.: The Ramsey number R(3,t) has order of magnitude t 2/log t. Random Struct. Algorithms 7(3), 173–207 (1995)MATHCrossRefGoogle Scholar
  8. 8.
    Liu, G.: Minimum clique number, chromatic number, and Ramsey numbers. Electron. J. Comb. 19, Research Paper 55, 10 (electronic, 2012). http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i1p55
  9. 9.
    Lovász, L., Plummer, M.D.: Matching theory, North-Holland Mathematics Studies, vol. 121. Annals of Discrete Mathematics, vol. 29. North-Holland Publishing Co., Amsterdam (1986)Google Scholar
  10. 10.
    Radziszowski, S.P.: Small Ramsey numbers. Electron. J. Comb. 1, Dynamic Survey 1, (electronic, 1994). http://www.combinatorics.org/Surveys/index.html. (2011)
  11. 11.
    Xu X.D., Xie Z., Radziszowski S.P.: A constructive approach for the lower bounds on the Ramsey numbers R(s,t). J. Graph Theory 47(3), 231–239 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer 2012

Authors and Affiliations

  • Csaba Biró
    • 1
  • Zoltán Füredi
    • 2
    • 3
  • Sogol Jahanbekam
    • 2
  1. 1.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Rényi Institute of Mathematics of the Hungarian Academy of SciencesBudapestHungary

Personalised recommendations