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arXiv · 2102.10220

Making an $H$-Free Graph $k$-Colorable

Jacob Fox·Zoe Himwich·Nitya Mani
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Abstract

We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For $H$ any fixed odd cycle, we determine the answer up to a constant factor when $n$ is sufficiently large. We also prove an upper bound when $H$ is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges.

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Jacob Fox, Zoe Himwich, Nitya Mani. 2021-02-20. Making an $H$-Free Graph $k$-Colorable. https://arxiv.org/abs/2102.10220

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