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

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

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    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.

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Correspondence to Maria Chudnovsky.

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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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Mathematics Subject Classification (2000)

  • 05C15
  • 05C17