A survey of χ‐boundedness

Scott, Alex; Seymour, Paul

doi:10.1002/jgt.22601

Cited by 138 publications

(90 citation statements)

References 114 publications

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“…Gyárfás [12] conjectured that the converse is also true (known as the Gyárfás Conjecture), and proved the conjecture when

H = P_{t}

[13]: every

P_{t}

‐free graph

G

has

χ (G) \leq {MathClass-open(t - 1 MathClass-close)}^{ω (G) - 1}

. Similar to results in [21], this

χ

‐binding function is exponential in

ω (G)

. It is natural to ask the following question.…”

Section: Introductionmentioning

confidence: 86%

“…In recent years, there has been an ongoing project led by Scott and Seymour that aims to determine the existence of

χ

‐binding functions for classes of graphs without holes of various lengths. We refer the reader to the recent survey by Scott and Seymour [21] for various nice results. One thing to note is that most

χ

‐binding functions in this setting are exponential.…”

Section: Introductionmentioning

confidence: 99%

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An optimal χ‐bound for (P6, diamond)‐free graphs

Cameron

Huang

Merkel

2021

Journal of Graph Theory

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In this paper we show that every ( P 6, diamond)‐free graph G satisfies χ ( G ) ≤ ω ( G ) + 3, where χ ( G ) and ω ( G ) are the chromatic number and clique number of G, respectively. Our bound is attained by the complement of the famous 27‐vertex Schläfli graph. Our result unifies previously known results on the existence of linear χ‐binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect ( P 6, diamond)‐free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well‐known characterization of 3‐colourable ( P 6 , K 3 )‐free graphs.

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“…Gyárfás [12] conjectured that the converse is also true (known as the Gyárfás Conjecture), and proved the conjecture when

H = P_{t}

[13]: every

P_{t}

‐free graph

G

has

χ (G) \leq {MathClass-open(t - 1 MathClass-close)}^{ω (G) - 1}

. Similar to results in [21], this

χ

‐binding function is exponential in

ω (G)

. It is natural to ask the following question.…”

Section: Introductionmentioning

confidence: 86%

“…In recent years, there has been an ongoing project led by Scott and Seymour that aims to determine the existence of

χ

χ

‐binding functions in this setting are exponential.…”

Section: Introductionmentioning

confidence: 99%

An optimal χ‐bound for (P6, diamond)‐free graphs

Cameron

Huang

Merkel

2021

Journal of Graph Theory

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“…Thus, for instance, the class of

C_{3}

‐free (or triangle‐free) graphs is not

χ

‐bounded. We refer to [16, 17] for more results on

χ

‐bounded classes of

MJX-tex-caligraphicscriptF

‐free graphs, and we give below some of them which are related to our results.…”

Section: Introductionmentioning

confidence: 82%

Coloring graphs with no induced five‐vertex path or gem

Chudnovsky

Karthick

Maceli

et al. 2020

Journal of Graph Theory

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“…Moreover, Erdős's celebrated result on the existence of graphs of high girth and high chromatic number [12] implies that if F is finite, then at least one member of F must be a forest. These two facts constitute the "only if" part of the following tantalizing and still widely open conjecture of Gyárfás and Sumner (see [28] for a survey on known results).…”

Section: Introductionmentioning

confidence: 84%

Extension of Gyárfás-Sumner Conjecture to Digraphs

Aboulker¹,

Charbit

Naserasr³

2021

Electron. J. Combin.

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The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph. Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

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A survey of χ‐boundedness

Cited by 138 publications

References 114 publications

An optimal χ‐bound for (P6, diamond)‐free graphs

An optimal χ‐bound for (P6, diamond)‐free graphs

Coloring graphs with no induced five‐vertex path or gem

Extension of Gyárfás-Sumner Conjecture to Digraphs

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