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Polynomial $χ$-boundedness for excluding $P_5$

Tung H. Nguyen

TL;DR

This work settles the polynomial $\chi$-boundedness question for the forbidden induced subgraph $P_5$ by proving there exists $d\ge2$ with $\chi(G) \le \omega(G)^d$ for all $P_5$-free graphs $G$, resolving a 1985 Gyárfás problem. The authors develop a triad of methods: (i) a chromatic-analogue Rödl-type structure for $P_5$-free graphs based on Gyárfás path ideas; (ii) a decomposition along high-$\chi$ anticomplete pairs that either yields large complete blocks or polynomially dense induced subgraphs; and (iii) a two-round chromatic density increment leveraging the Erdős–Hajnal theorem for $P_5$. Together these yield a robust framework that converts density-type information into global bounds on $\chi(G)$ in terms of $\omega(G)$. The results advance the broader program of polynomially bounding chromatic number in hereditary classes and illustrate how chromatic-density concepts can substitute counting techniques in a purely chromatic setting. The approach has potential implications for related forest-forbidden classes and motivates further refinement toward tighter exponents and broader forest families.

Abstract

We obtain some $d\ge2$ such that every graph $G$ with no induced copy of the five-vertex path $P_5$ has chromatic number at most $ω(G)^d$, thereby resolving an open problem of Gyárfás from 1985. The proof consists of three main ingredients: $\bullet$ an analogue of Rödl's theorem for the chromatic number of $P_5$-free graphs, proved via the ``Gyárfás path'' argument; $\bullet$ a decomposition argument for $P_5$-free graphs that allows one to grow high-chromatic anticomplete pairs indefinitely or to capture a polynomially chromatic-dense induced subgraph; and $\bullet$ a ``chromatic density increment'' argument that uses the Erdős-Hajnal result for $P_5$ as a black box.

Polynomial $χ$-boundedness for excluding $P_5$

TL;DR

This work settles the polynomial -boundedness question for the forbidden induced subgraph by proving there exists with for all -free graphs , resolving a 1985 Gyárfás problem. The authors develop a triad of methods: (i) a chromatic-analogue Rödl-type structure for -free graphs based on Gyárfás path ideas; (ii) a decomposition along high- anticomplete pairs that either yields large complete blocks or polynomially dense induced subgraphs; and (iii) a two-round chromatic density increment leveraging the Erdős–Hajnal theorem for . Together these yield a robust framework that converts density-type information into global bounds on in terms of . The results advance the broader program of polynomially bounding chromatic number in hereditary classes and illustrate how chromatic-density concepts can substitute counting techniques in a purely chromatic setting. The approach has potential implications for related forest-forbidden classes and motivates further refinement toward tighter exponents and broader forest families.

Abstract

We obtain some such that every graph with no induced copy of the five-vertex path has chromatic number at most , thereby resolving an open problem of Gyárfás from 1985. The proof consists of three main ingredients: an analogue of Rödl's theorem for the chromatic number of -free graphs, proved via the ``Gyárfás path'' argument; a decomposition argument for -free graphs that allows one to grow high-chromatic anticomplete pairs indefinitely or to capture a polynomially chromatic-dense induced subgraph; and a ``chromatic density increment'' argument that uses the Erdős-Hajnal result for as a black box.
Paper Structure (13 sections, 34 theorems, 35 equations, 5 figures)

This paper contains 13 sections, 34 theorems, 35 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Background and discussion
  4. Preliminaries and some proof ideas
  5. Chromatic analogues of Rödl's theorem
  6. Decomposing along high-$\chi$ anticomplete pairs
  7. Chromatic density increment
  8. Linear-$\chi$ pure pairs versus linearly $\chi$-dense induced subgraphs
  9. Growing high-$\chi$ anticomplete pairs or capturing $\chi$-dense subgraphs
  10. Chromatic density increment, first round
  11. Anticomplete or $\chi$-dense blockades via Erdős--Hajnal
  12. Chromatic density increment, second round
  13. Additional remarks

Key Result

Theorem 1.3

There exists $d\ge2$ such that every $P_5$-free graph $G$ satisfies $\chi(G)\le\omega(G)^{d}$.

Figures (5)

  • Figure 1: Proof of \ref{['claim:bip']}.
  • Figure 2: The decomposition and proof of \ref{['claim:2k2']}.
  • Figure 3: Proof of \ref{['claim:smalls']}; the ellipse represents $S\setminus N_G[v]$.
  • Figure 4: Proof of \ref{['claim:avgp52']}.
  • Figure 5: Proof of \ref{['lem:2dense1']}.

Theorems & Definitions (58)

  • Conjecture 1.1: Gyárfás--Sumner
  • Conjecture 1.2: Polynomial Gyárfás--Sumner
  • Theorem 1.3
  • Conjecture 1.4: Erdős--Hajnal
  • Theorem 2.1
  • Theorem 2.2: Rödl
  • Theorem 2.3: Chudnovsky--Scott--Seymour--Spirkl
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 48 more