Polynomial $\chi$-boundedness for excluding $P_5$

TL;DR

This paper proves that for graphs excluding an induced $P_5$, the chromatic number is polynomially bounded by the clique number, resolving a long-standing open problem from 1985.

Contribution

It introduces a novel chromatic density framework and leverages the Erdős-Hajnal result to establish polynomial $ ext{χ}$-boundedness for $P_5$-free graphs.

Findings

Chromatic number is polynomially bounded by clique number for $P_5$-free graphs.

Introduces a new chromatic density framework involving quasirandomness and density increment.

Connects the problem to the Erdős-Hajnal result for $P_5$.

Abstract

Resolving a 1985 open problem of Gy\'arf\'as, we prove that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path $P_5$. Our approach introduces a chromatic density framework involving chromatic quasirandomness and chromatic density increment, which allows us to deduce the desired statement from the Erd\H{o}s-Hajnal result for $P_5$.