Induced Subgraphs of Graphs With Large Chromatic Number. X. Holes of Specific Residue

Abstract

A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of previous papers, showing that, for every positive integer k, every graph with large chromatic number contains either a large complete subgraph or induced cycles of all lengths modulo k. As an application, we prove two conjectures of Kalai and Meshulam from the 1990’s connecting the chromatic number of a graph with the homology of its independence complex.

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Acknowledgement

We would like to thank Gil Kalai for helpful discussions.

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Correspondence to Alex Scott.

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Supported by a Leverhulme Trust Research Fellowship.

Supported by ONR grant N00014-14-1-0084, NSF grant DMS-1265563, NSF grant DMS-1800053 and AFOSR grant A9550-19-1-0187.

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Scott, A., Seymour, P. Induced Subgraphs of Graphs With Large Chromatic Number. X. Holes of Specific Residue. Combinatorica 39, 1105–1132 (2019). https://doi.org/10.1007/s00493-019-3804-y

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Mathematics Subject Classification (2010)

  • 05C15
  • 05C39
  • 05C75