Progress on the Borodin–Kostochka conjecture: A structural approach via vertex partitions relative to a maximum clique

The Borodin-Kostochka conjecture states that for any graph $G$ with $\mathrm{\Delta }(G)\ge 9$, we have $\chi (G)\le max\{\mathrm{\Delta }(G)-1,\omega (G)\}$. In this paper, we study the structure of potential counterexamples by partitioning vertices according to the number of neighbors they have in a fixed maximum clique. This approach provides a sufficient condition for $\chi (G)\le \mathrm{\Delta }(G)-1$. Consequently, we confirm the conjecture for any $\overline{K_{1,t}}$-free graph $G$ with $t\ge 3$ and $\mathrm{\Delta }(G)\ge 2t+1$, strengthening and extending the recent work of Lan and Lin in 2024.