Sharp linear Tur\'an remainders for cycle and edge extremality

Abstract

For a graph $G$, let $e(G)$ denote the number of edges of $G$, and let $c(G)$ be the number of distinct cycles in $G$. Morrison, Roberts and Scott asked whether, for every fixed graph $H$ and all large $n$, some $n$-vertex $H$-free graph maximizes both $e(G)$ and $c(G)$. In this paper, we show that the answer is no for every possible chromatic number. Let $T_{n,r}$ be the complete $r$-partite Tur\'an graph on $n$ vertices. We find a constant $\gamma_r>0$ such that if $n$ is sufficiently large and $\mathcal{H}$ is a finite family with $\chi(\mathcal{H})=r+1$ satisfying \[ ex(n,\mathcal{H})<e(T_{n,r})+\gamma_r n, \] then every cycle-maximal $\mathcal{H}$-free graph is edge-extremal, and $\gamma_r$ is best possible.