Dense Induced Bipartite Subgraphs in Triangle-Free Graphs

Abstract

The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least cH log d/log log d, thus nearly confirming one and proving another conjecture of Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdœs, Janson, Łuczak and Spencer.

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Acknowledgements

We would like to thank the anonymous referees for their helpful comments and suggestions.

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Correspondence to Shoham Letzter.

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Research supported in part by SNSF project 178493.

Research supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation.

Research supported in part by SNSF grant 200021-175573.

Research supported by the Alexander von Humboldt Foundation.

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Kwan, M., Letzter, S., Sudakov, B. et al. Dense Induced Bipartite Subgraphs in Triangle-Free Graphs. Combinatorica 40, 283–305 (2020). https://doi.org/10.1007/s00493-019-4086-0

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Mathematics Subject Classification (2010)

  • 05C35