Coloring graphs with no long induced path

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 $G$. 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 lower the base of the exponential by one: for every $t\ge5$, every $P_t$-free graph $G$ satisfies \[ \chi(G)\le 3\,(t-3)^{\omega(G)+4}. \] The proof combines two refinements of the Gy\'arf\'as path argument and was developed with the assistance of Claude Fable 5.1 of Anthropic and GPT Pro of OpenAI.