Borodin-Kostochka conjecture for a family of $P_6$-free graphs

TL;DR

The paper addresses whether graphs with large maximum degree within the ($P_6$, apple, torch)-free hereditary class satisfy $\chi(G)\le \max\{\Delta(G)-1,\omega(G)\}$. It develops a minimal-counterexample framework based on a $(u,\phi)$-coloring and a relaxed-graph encoding to capture color-missing structures, enabling reduction to $\Delta=9$ and vertex-critical cases. The main result is proved: if $G$ is ($P_6$, apple, torch)-free and $\Delta(G)\ge 9$, then $\chi(G)\le \max\{\Delta(G)-1,\omega(G)\}$. The authors also discuss connections to odd-hole-free graphs and outline open questions for related six-vertex forbidden subgraph families, highlighting the broader applicability of their structural approach.

Abstract

Borodin and Kostochka conjectured that every graph $G$ with $Δ\ge9$ satisfies $χ\le$ max $\{ω, Δ-1\}$. Gupta and Pradhan proved the Borodin-Kostochka conjecture for ($P_5$, $C_4$)-free graphs [{\em J. Appl. Math. Comp.} \textbf{65} (2021) 877-884]. In this paper, we prove the Borodin-Kostochka conjecture for ($P_6$, apple, torch)-free graphs, that is, graphs with no induced $P_6$, no induced $C_5$ with a hanging edge, and no induced $C_5$ and $C_4$ sharing exactly an induced $P_3$. This generalizes the result of Gupta and Pradhan from the perspective of allowing the existence of $P_5$.

Figures (1)

• Figure 1: Illustration of chair, apple and torch.

Theorems & Definitions (10)

• Theorem 1.1
• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.1
• Claim 3.1
• Claim 3.2