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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 graphtheory are special cases of this question. For $H$ any fixed odd cycle, wedetermine the answer up to a constant factor when $n$ is sufficiently large. Wealso prove an upper bound when $H$ is a fixed clique that we conjecture istight up to a constant factor, and prove upper bounds for more general familiesof graphs. We apply our results to get a new bound on the maximum cut of graphswith a forbidden odd cycle in terms of the number of edges.

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