Polynomial $\chi$-boundedness for excluding $P_5$
Combinatorics
2026-05-12 v3
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 . 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 .
Keywords
Cite
@article{arxiv.2512.24907,
title = {Polynomial $\chi$-boundedness for excluding $P_5$},
author = {Tung H. Nguyen},
journal= {arXiv preprint arXiv:2512.24907},
year = {2026}
}
Comments
v3: 40 pages plus bibliography and appendices, supersedes arXiv:2504.21127 and arXiv:2510.05724. Minor inaccuracies from v2 corrected