On the minimum degree forcing F-free graphs to be (nearly) bipartite

Abstract

Let $\beta (G)$ denote the minimum number of edges to be removed from a graph G to make it bipartite. For each 3-chromatic graph F we determine a parameter $\xi (F)$ such that for each F-free graph G on n vertices with minimum degree $\delta (G)⩾2n/(\xi (F)+2)+\mathrm{o}(n)$ we have $\beta (G)=\mathrm{o}(n^2)$, while there are F-free graphs H with $\delta (H)\ge \lfloor 2n/(\xi (F)+2)\rfloor$ for which $\beta (H)=\Omega (n^2)$.