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arXiv · 2606.07358

Polynomial minimum degree stability for $3$-chromatic graphs

Yisai Xue
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Abstract

Let $H$ be a fixed 3-chromatic graph, and let $g \geq 2$ be the smallest integer such that there is no homomorphism $H \rightarrow C_{2 g+1}$. For every $\varepsilon>0$, we prove that there exists a constant $\rho>0$ such that every $H$-free $n$-vertex graph $G$ with $\delta(G) \geq(2 /(2 g+1)+\varepsilon) n$ can be made bipartite by deleting $O\left(n^{2-\rho}\right)$ edges. Thus the sharp qualitative minimum-degree stability theorem for 3-chromatic graphs admits a polynomial strengthening. In particular, this gives an affirmative answer to a question of Illingworth [\textit{Minimum degree stability of $H$-free graphs}, Combinatorica, 43(1):129-147, 2023.] on blow-ups of odd cycles.

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Yisai Xue. 2026-06-05. Polynomial minimum degree stability for $3$-chromatic graphs. https://arxiv.org/abs/2606.07358

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