Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

Alex Scott; Paul Seymour; Sophie Spirkl

doi:10.1007/s00493-023-00015-w

Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

Alex Scott, Paul Seymour, Sophie Spirkl

Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

A graph G is H -free if it has no induced subgraph isomorphic to H. We prove that a P₅ -free graph with clique number ω≥ 3 has chromatic number at most ωlog2(ω) . The best previous result was an exponential upper bound (5 / 27) 3 ^ω , due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for P₅ , which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P₅ -free graphs, and our result is an attempt to approach that.

Original language	English (US)
Pages (from-to)	845-852
Number of pages	8
Journal	Combinatorica
Volume	43
Issue number	5
DOIs	https://doi.org/10.1007/s00493-023-00015-w
State	Published - Oct 2023

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics
Computational Mathematics

Keywords

Chromatic number
Induced subgraphs

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10.1007/s00493-023-00015-w

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title = "Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path",

abstract = "A graph G is H -free if it has no induced subgraph isomorphic to H. We prove that a P5 -free graph with clique number ω≥ 3 has chromatic number at most ωlog2(ω) . The best previous result was an exponential upper bound (5 / 27) 3 ω , due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erd{\H o}s-Hajnal conjecture holds for P5 , which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P5 -free graphs, and our result is an attempt to approach that.",

keywords = "Chromatic number, Induced subgraphs",

author = "Alex Scott and Paul Seymour and Sophie Spirkl",

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year = "2023",

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doi = "10.1007/s00493-023-00015-w",

language = "English (US)",

volume = "43",

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TY - JOUR

T1 - Polynomial Bounds for Chromatic Number. IV

T2 - A Near-polynomial Bound for Excluding the Five-vertex Path

AU - Scott, Alex

AU - Seymour, Paul

AU - Spirkl, Sophie

PY - 2023/10

Y1 - 2023/10

N2 - A graph G is H -free if it has no induced subgraph isomorphic to H. We prove that a P5 -free graph with clique number ω≥ 3 has chromatic number at most ωlog2(ω) . The best previous result was an exponential upper bound (5 / 27) 3 ω , due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for P5 , which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P5 -free graphs, and our result is an attempt to approach that.

AB - A graph G is H -free if it has no induced subgraph isomorphic to H. We prove that a P5 -free graph with clique number ω≥ 3 has chromatic number at most ωlog2(ω) . The best previous result was an exponential upper bound (5 / 27) 3 ω , due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for P5 , which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for P5 -free graphs, and our result is an attempt to approach that.

KW - Chromatic number

KW - Induced subgraphs

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JO - Combinatorica

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Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path

Abstract

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