Making an H $H$‐free graph k $k$‐colorable. Issue 2 (16th August 2022) - Fox, Jacob
Abstract: We study the following question: How few edges can we delete from any H $H$ ‐free graph on n $n$ vertices to make the resulting graph k $k$ ‐colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For H $H$ any fixed odd cycle, we determine the answer up to a constant factor when n $n$ is sufficiently large. We also prove an upper bound when H $H$ 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.