Graphs and Combinatorics

, Volume 1, Issue 1, pp 91–96 | Cite as

On graphs with small subgraphs of large chromatic number

  • V. Rödl
  • R. A. Duke


Bollobás, Erdös, Simonovits, and Szemerédi conjectured [1] that for each positive constantc there exists a constantg(c) such that ifG is any graph which cannot be made 3-chromatic by the omission ofcn2 edges, thenG contains a 4-chromatic subgraph with at mostg(c) vertices. Here we establish the following generalization which was suggested by Erdös [2]: For each positive constantc and positive integerk there exist positive integersfk(c) andno such that ifG is any graph with more thanno vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn2 edges, thenG contains ak-chromatic subgraph with at mostfk(c) vertices.


Positive Integer Independent Random Variable Chromatic Number Trinity College Extremal Graph 
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  1. 1.
    Bollobás, B., Erdös, P., Simonovits, M., Szemerédi, E.: Extremal graphs without large forbidden subgraphs. In: Advances in Graph Theory (Cambridge Comb. Conf. Trinity College, 1977). Ann. Discrete Math.3, 29–41 (1978)Google Scholar
  2. 2.
    Erdös, P.: Problems and results of graphs and hypergraphs: similarities and differences. In: Recent Progress in Ramsey Theory, edited by Nešetřil, J., Rödl, V. (Proc. Third Czechoslovak Symposium on Graph Theory, Prague 1982)Google Scholar
  3. 3.
    Szemerédi, E.: Regular partitions of graphs. Proc. Colloq. Int. C.N.R.S., pp. 399–401 (1976)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • V. Rödl
    • 1
  • R. A. Duke
    • 2
  1. 1.F.J.F.I., C.V.U.T.Praha 1Czechoslovakia
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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