Coloring graphs with no long induced path

2026-09-08

AI Summary

该研究解决了无长诱导路径图的着色问题，通过改进Gyárfás路径论证方法，得到了更优的着色数上界。

Abstract

Let $P_t$ denote the induced path on $t$ vertices. Let $\omega(G)$ denote the maximum number of vertices in a clique of a graph $G$. Previously Gy\'arf\'as (1987) proved that every $P_t$-free graph $G$ satisfies $\chi(G)\le(t-1)^{\omega(G)-1}$, and Gravier, Ho\`ang, and Maffray (2003) improved this to $\chi(G)\le (t-2)^{\omega(G)-1}$ for $t\ge4$. We prove that for $t\ge5$,every $P_t$-free graph $G$ satisfies $\chi(G)<c_t\lambda_t^{\omega(G)-1}$, where $\lambda_t=\tfrac12\bigl(t-2+\sqrt{t(t-4)}\bigr)<t-2$ and $c_t=\sqrt{1+4/(t(t-4))}=1+O(t^{-2})$ as $t\to\infty$. The proof is based on a refinement of the Gy\'arf\'as path argument and was found by Claude Fable 5.1 of Anthropic.

Problem

• Coloring
• Graphs
• Induced Path
• Clique

Innovation

• P_t-free graph
• chromatic number
• clique number
• refinement of Gyárfás path argument