On Borodin-Kostochka conjecture for correspondence coloring

TL;DR

This paper proves a strengthened version of the Borodin-Kostochka conjecture for the correspondence chromatic number, showing that for large maximum degree, the correspondence chromatic number is bounded by the maximum of the clique number and degree minus one.

Contribution

It extends the Borodin-Kostochka conjecture to the correspondence chromatic number for graphs with sufficiently large maximum degree, improving upon previous results for standard and list chromatic numbers.

Findings

Proves the conjecture for correspondence chromatic number when maximum degree ≥ 3×10^9.

Shows that χ_{DP}(G) ≤ max(ω(G), Δ-1) for large Δ.

Strengthens earlier bounds for list and usual chromatic numbers.

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.