Making an H $H$‐free graph k $k$‐colorable

Abstract

AbstractWe study the following question: How few edges can we delete from any ‐free graph on vertices to make the resulting graph ‐colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For any fixed odd cycle, we determine the answer up to a constant factor when is sufficiently large. We also prove an upper bound when 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.