  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.
