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.

翻译：我们研究下面的问题:我们如何从任何无$的图表中删除少数边缘,以便从任何无$的图表中删除美元顶端,从而使所产生的图形能变色?结果显示,极端图形理论中的各种古典问题都是这一问题的特殊案例。对于任何固定的奇数周期,只要美元足够大,我们就能确定一个恒定系数。当美元是一个固定的俱乐部,我们推测美元会紧凑到一个恒定系数,并证明对更普通的图表家族来说是上界。我们运用我们的结果,在以奇数为底数的被禁止奇数周期的图表的最大截面上找到一个新的界限。