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arXiv · 2603.14427

On Borodin-Kostochka conjecture for correspondence coloring

Zdeněk Dvořák·Ross J. Kang·David Mikšaník
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Abstract

Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $\Delta(G)\ge 9$ and its clique number satisfies $\omega(G)\le \Delta(G)-1$, then its chromatic number satisfies $\chi(G) \le \Delta(G)-1$. We prove this statement with respect to a stronger graph coloring parameter, the correspondence chromatic number $\chi_{DP}$, provided the maximum degree is sufficiently large. More precisely, we prove that for every integer $\Delta\ge 3\cdot 10^9$, a graph $G$ of maximum degree at most $\Delta$ satisfies $\chi_{DP}(G) \le \max(\omega(G),\Delta-1)$. This strengthens earlier results of Reed (1999) for usual chromatic number and of Choi, Kierstead and Rabern (2023) for list chromatic number.

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Zdeněk Dvořák, Ross J. Kang, David Mikšaník. 2026-03-15. On Borodin-Kostochka conjecture for correspondence coloring. https://arxiv.org/abs/2603.14427

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