On the Borodin--Kostochka conjecture for graphs with large maximum degree

TL;DR

This paper verifies the Borodin--Kostochka conjecture for graphs with very large maximum degree, specifically proving the chromatic number bound for degrees above approximately 5.3 million, improving previous results.

Contribution

It proves the conjecture for graphs with sufficiently large maximum degree, specifically for degrees over 5.3 million, advancing the understanding of graph coloring.

Findings

Graphs with maximum degree ≥ 5.3 million and clique number less than degree satisfy χ(G) ≤ Δ - 1.

The result confirms the conjecture for extremely large degrees, surpassing previous bounds.

Improves a longstanding result of Reed on the conjecture.

Abstract

The Borodin--Kostochka conjecture states that every graph $G$ with maximum degree $\Delta(G)\ge 9$ satisfies $\chi(G)\le \max\{\omega(G),\Delta(G)-1\}$. In this paper, we verify this conjecture for graphs with sufficiently large maximum degree. More precisely, we prove that every graph $G$ with maximum degree $\Delta \ge 5.3\times 10^6$ and clique number $\omega(G)<\Delta$ satisfies $\chi(G)\le \Delta-1$. This improves a longstanding result of Reed.