Minimum degree stability for graphs without odd-cycle blow-up

Abstract

For fixed integers $g\ge 2$ and $t\ge 1$, and every $\epsilon >0$, we prove that there exists a constant $\ddot{I}>0$ such that every $n$-vertex graph $G$ with $\hat{I}´(G)\ge (2/(2g+1)+\epsilon )n$ either contains $C_{2g-1}[t]$, or can be made bipartite by deleting $O(n^{2-\ddot{I}})$ edges. This gives an affirmative answer to a question of Illingworth in [Minimum degree stability of $H$-free graphs, Combinatorica, 43(1):129-147, 2023.]

8 pages