Cycles in triangle-free graphs of large chromatic number

Abstract

More than twenty years ago Erdős conjectured [4] that a triangle-free graph G of chromatic number kk 0(ε) contains cycles of at least k 2−ε different lengths as k→∞. In this paper, we prove the stronger fact that every triangle-free graph G of chromatic number kk 0(ε) contains cycles of 1/64(1 − ε)k 2 logk/4 consecutive lengths, and a cycle of length at least 1/4(1 − ε)k 2logk. As there exist triangle-free graphs of chromatic number k with at most roughly 4k 2 logk vertices for large k, these results are tight up to a constant factor. We also give new lower bounds on the circumference and the number of different cycle lengths for k-chromatic graphs in other monotone classes, in particular, for K r -free graphs and graphs without odd cycles C 2s+1.

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

References

  1. [1]

    M. Ajtai, J. Komlós and E. Szemerédi: A note on Ramsey numbers, J. Combin. Theory Ser. A 29 (1980), 354–360.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    T. Bohman and P. Keevash: Dynamic concentration of the triangle-free process, http://arxiv.org/abs/1302.5963.

  3. [3]

    J. A. Bondy and M. Simonovits: Cycles of even length in graphs, J. Combin. Theory Ser. B 16 (1974), 97–105.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    P. Erdős: Some of my favourite problems in various branches of combinatorics, Matematiche (Catania) 47 (1992), 231–240.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    G. Fiz Pontiveros, S. Griffiths and R. Morris: The triangle-free process and R(3;k), http://arxiv.org/abs/1302.6279.

  6. [6]

    A. Gyárfás: Graphs with k odd cycle lengths, Discrete Mathematics 103 (1992), 41–48.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    T. R. Jensen and B. Toft: Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, (1995).

    Google Scholar 

  8. [8]

    J. Kim: The Ramsey number R(3; t) has order of magnitude t 2=log t, Random Structures and Algorithms 7 (1995), 173–207.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    P. Mihók and I. Schiermeyer: Cycle lengths and chromatic number of graphs, Discrete Math. 286 (2004), 147–149.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    J. Shearer: A note on the independence number of triangle-free graphs, Discrete Mathematics 46 (1983), 83–87.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    B. Sudakov: A note on odd cycle-complete graph Ramsey numbers, Electron. J. of Combin. 9 (2002).

    Google Scholar 

  12. [12]

    B. Sudakov and J. Verstraete: Cycle lengths in sparse graphs, Combinatorica 28 (2008), 357–372.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    B. Sudakov and J. Verstraete: Cycles in graphs with large independence ratio, Journal of Combinatorics 2 (2011), 82–102.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    J. Verstraete: Arithmetic progressions of cycle lengths in graphs, Combinatorics, Probability and Computing 9 (2000), 369–373.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alexandr Kostochka.

Additional information

The authors thank Institute Mittag-Leffler (Djursholm, Sweden) for the hospitality and creative environment.

Research supported in part by NSF grant DMS-1266016 and by Grant NSh.1939.2014.1 of the President of Russia for Leading Scientific Schools.

Research supported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant.

Research supported by NSF Grant DMS-1362650.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kostochka, A., Sudakov, B. & Verstraëte, J. Cycles in triangle-free graphs of large chromatic number. Combinatorica 37, 481–494 (2017). https://doi.org/10.1007/s00493-015-3262-0

Download citation

Mathematics Subject Classification (2000)

  • 05C15
  • 05C35
  • 05C38