# Making an H-Free Graph k-Colorable

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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