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Induced Subgraphs of Graphs with Large Chromatic Number. III. Long Holes

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Abstract

We prove a 1985 conjecture of Gyárfás that for all k, ℓ, every graph with sufficiently large chromatic number contains either a clique of cardinality more than k or an induced cycle of length more than ℓ.

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References

  1. M. Chudnovsky, A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyárfás’ conjectures, J. Combina-torial Theory, Ser. B, to appear, arXiv:1411.6465.

  2. A. Gyárfás: Problems from the world surrounding perfect graphs, Proceedings of the International Conference on Combinatorial Analysis and its Applications, (Pokrzywna, 1985), Zastos. Mat. 19 (1987), 413–441.

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  3. A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. I. Odd holes, J. Combinatorial Theory, Ser. B, to appear, arXiv:1410.4118.

  4. A. Scott and P. Seymour: Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes, submitted for publication (manuscript January 2015), arXiv:1509.06563.

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Authors and Affiliations

  1. Columbia University, New York, NY, 10027, USA

    Maria Chudnovsky

  2. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Alex Scott

  3. Princeton University, Princeton, NJ, 08544, USA

    Paul Seymour

Authors
  1. Maria Chudnovsky
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  2. Alex Scott
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  3. Paul Seymour
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Corresponding author

Correspondence to Maria Chudnovsky.

Additional information

Supported by NSF grant DMS-1265803.

Supported by ONR grant N00014-14-1-0084 and NSF grant DMS-1265563.

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Chudnovsky, M., Scott, A. & Seymour, P. Induced Subgraphs of Graphs with Large Chromatic Number. III. Long Holes. Combinatorica 37, 1057–1072 (2017). https://doi.org/10.1007/s00493-016-3467-x

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  • DOI: https://doi.org/10.1007/s00493-016-3467-x

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