On Borodin-Kostochka conjecture for correspondence coloring

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.

arXiv comment31 pages, 1 figure