Stability with minuscule structure for chromatic thresholds

TL;DR

This paper establishes stability theorems for graphs near the chromatic threshold, revealing their structural closeness to extremal configurations and providing detailed insights especially when the forbidden subgraph is a clique.

Contribution

It introduces the first stability results capturing lower-order structural features of extremal graphs near the chromatic threshold, including for fractional and bounded-VC variants.

Findings

Graphs near the chromatic threshold are structurally close to extremal configurations.

For clique-free graphs, a partition into independent sets and a small subgraph with fractional chromatic number close to 2 is possible.

Determined fractional and bounded-VC chromatic thresholds for all graphs and cliques, respectively.

Abstract

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $d n$ is bounded by a constant depending only on $H$ and $d$. Allen, B{\"o}ttcher, Griffiths, Kohayakawa, and Morris determined the chromatic threshold for every $H$; in particular, they showed that if $\chi(H)=r\ge 3$, then $\delta_\chi(H) \in\{\frac{r-3}{r-2},~\frac{2 r-5}{2 r-3},~\frac{r-2}{r-1}\}$. While the chromatic thresholds have been completely determined, rather surprisingly the structural behaviors of extremal graphs near the threshold remain unexplored. In this paper, we establish the stability theorems for chromatic threshold problems. We prove that every $n$-vertex $H$-free graph $G$ with $\delta(G)\ge (\delta_\chi(H)-o(1))n$ and $\chi(G)=\omega(1)$ must be structurally close to one of the extremal configurations. Furthermore, we give a stronger stability result when $H$ is a clique, showing that $G$ admits a partition into independent sets and a small subgraph on sublinear number of vertices. We show that this small subgraph has fractional chromatic number $2+o(1)$ and is homomorphic to a Kneser graph defined by subsets of a logarithmic size set; both these two bounds are best possible. This is the first stability result that captures the lower-order structural features of extremal graphs. We also study two variations of chromatic thresholds. Replacing chromatic number by its fractional counterpart, we determine the fractional chromatic thresholds for all graphs. Another variation is the bounded-VC chromatic thresholds, which was introduced by Liu, Shangguan, Skokan, and Xu very recently. Extending work of {\L}uczak and Thomass{\'e} on the triangle case, we determine the bounded-VC chromatic thresholds for all cliques.