Borodin–Kostochka Conjecture Holds for Odd-Hole-Free Graphs

The Borodin–Kostochka Conjecture states that for a graph G, if \(\Delta (G)\ge 9\) $\mathrm{\Delta }(G)\ge 9$ , then \(\chi (G)\le \max \{\Delta (G)-1,\omega (G)\}\) $\chi (G)\le max\{\mathrm{\Delta }(G)-1,\omega (G)\}$ . In this paper, we prove the Borodin–Kostochka Conjecture holding for odd-hole-free graphs.