Regular partitions of gentle graphs

Jiang, Yiting; Nešetřil, Jaroslav; Mendez, Patrice Ossona de; Siebertz, Sebastian

doi:10.1007/s10474-020-01074-x

Cited by 6 publications

(4 citation statements)

References 58 publications

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“…This proposition was already shown in [13,Theorem 50], but we propose an arguably simpler explanation of this fact.…”

Section: Introductionsupporting

confidence: 73%

Erdős-Hajnal properties for powers of sparse graphs

Briański,

Micek,

Pilipczuk

et al. 2020

Preprint

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We prove that for every nowhere dense class of graphs C, positive integer d, and ε > 0, the following holds: in every n-vertex graph G from C one can find two disjoint vertex subsets A, B ⊆ V (G) such that • |A| (1/2 − ε) • n and |B| = Ω(n 1−ε ); and • either dist(a, b) d for all a ∈ A and b ∈ B, or dist(a, b) > d for all a ∈ A and b ∈ B.We also show some stronger variants of this statement, including a generalization to the setting of First-Order interpretations of nowhere dense graph classes.

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“…This proposition was already shown in [13,Theorem 50], but we propose an arguably simpler explanation of this fact.…”

Section: Introductionsupporting

confidence: 73%

Erdős-Hajnal properties for powers of sparse graphs

Briański,

Micek,

Pilipczuk

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Note that obviously bω(G) ≥ ω(G)/2 . A class C with bω(C ) < ∞ is said to be weakly sparse [19] or biclique free.…”

Section: Definitions and Notationsmentioning

confidence: 99%

From χ- to χ-bounded classes

Jiang

Nešetřil

Mendez

2023

Journal of Combinatorial Theory, Series B

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χ-bounded classes are studied here in the context of star colorings and, more generally, χ pcolorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ 2 ) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χ p -boundedness leads to more stability and we give structural characterizations of (strong and weak) χ p -bounded classes. We also generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g > 3 even hole-free graphs G contain at most ϕ(g, ω(G)) |G| holes of length g.

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“…Note that obviously bω(G) ≥ ω(G)/2 . A class C with bω(C ) < ∞ is said to be weakly sparse [15] or biclique free.…”

Section: Definitions and Notationsmentioning

confidence: 99%

From $χ$- to $χ_p$-bounded classes

Jiang¹,

Nešetřil²,

Mendez³

2020

Preprint

Self Cite

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χ-bounded classes are studied here in the context of star colorings and more generally χ pcolorings. This leads to natural extensions of the notion of bounded expansion class and to structural characterization of these. In this paper we solve two conjectures related to star coloring boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. We give structural characterizations of (strong and weak) χ p -bounded classes. On the way, we generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g > 3 even hole-free graphs G contain at most ϕ(g, ω(G)) |G| holes of length g.

show abstract

Regular partitions of gentle graphs

Cited by 6 publications

References 58 publications

Erdős-Hajnal properties for powers of sparse graphs

Erdős-Hajnal properties for powers of sparse graphs

From χ- to χ-bounded classes

From $χ$- to $χ_p$-bounded classes

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