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

TL;DR

This paper investigates the minimum number of edges to delete from an $H$-free graph to make it $k$-colorable, providing bounds for various $H$ and applications to maximum cut problems.

Contribution

It offers new bounds and conjectures on edge deletions needed for $k$-colorability in $H$-free graphs, extending classical extremal graph theory results.

Findings

Determined bounds for odd cycle-free graphs when $n$ is large.

Proved an upper bound for clique-free graphs, conjectured to be tight.

Derived a new maximum cut bound for graphs with forbidden odd cycles.

Abstract

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.