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

The Borodin–Kostochka conjecture states that every graph $G$ with maximum degree $\mathrm{\Delta }(G)\ge 9$ satisfies $\chi (G)\le \max \{\omega (G),\mathrm{\Delta }(G)-1\}$. 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 $\mathrm{\Delta }\ge 5.3\times {10}^6$ and clique number $\omega (G)<\mathrm{\Delta }$ satisfies $\chi (G)\le \mathrm{\Delta }-1$. This improves a longstanding result of Reed.