The χ-Binding Function of d-Directional Segment Graphs.
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| Title: | The χ-Binding Function of d-Directional Segment Graphs. |
|---|---|
| Authors: | Duraj, Lech1 (AUTHOR), Kang, Ross J.2 (AUTHOR) r.kang@uva.nl, La, Hoang1 (AUTHOR) hoang.la.research@gmail.com, Narboni, Jonathan1 (AUTHOR), Pokrývka, Filip3 (AUTHOR) xpokryvk@fi.muni.cz, Rambaud, Clément4 (AUTHOR) clement.rambaud@inria.fr, Reinald, Amadeus5 (AUTHOR) amadeus.reinald@lirmm.fr |
| Source: | Discrete & Computational Geometry. Oct2025, Vol. 74 Issue 3, p758-770. 13p. |
| Subjects: | Intersection graph theory, Graph coloring, Mathematicians, Graph theory |
| Abstract: | Given a positive integer d, the class d-DIR is defined as all those intersection graphs formed from a finite collection of line segments in R 2 having at most d slopes. Since each slope induces an interval graph, it easily follows for every G in d-DIR with clique number at most ω that the chromatic number χ (G) of G is at most d ω . We show for every even value of ω how to construct a graph in d-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the χ -binding function of d-DIR is ω ↦ d ω for ω even and ω ↦ d (ω - 1) + 1 for ω odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case d = 2 . [ABSTRACT FROM AUTHOR] |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 188355826 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: The χ-Binding Function of d-Directional Segment Graphs. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Duraj%2C+Lech%22">Duraj, Lech</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Kang%2C+Ross+J%2E%22">Kang, Ross J.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> r.kang@uva.nl</i><br /><searchLink fieldCode="AR" term="%22La%2C+Hoang%22">La, Hoang</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> hoang.la.research@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Narboni%2C+Jonathan%22">Narboni, Jonathan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Pokrývka%2C+Filip%22">Pokrývka, Filip</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> xpokryvk@fi.muni.cz</i><br /><searchLink fieldCode="AR" term="%22Rambaud%2C+Clément%22">Rambaud, Clément</searchLink><relatesTo>4</relatesTo> (AUTHOR)<i> clement.rambaud@inria.fr</i><br /><searchLink fieldCode="AR" term="%22Reinald%2C+Amadeus%22">Reinald, Amadeus</searchLink><relatesTo>5</relatesTo> (AUTHOR)<i> amadeus.reinald@lirmm.fr</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Oct2025, Vol. 74 Issue 3, p758-770. 13p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Intersection+graph+theory%22">Intersection graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Graph+coloring%22">Graph coloring</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematicians%22">Mathematicians</searchLink><br /><searchLink fieldCode="DE" term="%22Graph+theory%22">Graph theory</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Given a positive integer d, the class d-DIR is defined as all those intersection graphs formed from a finite collection of line segments in R 2 having at most d slopes. Since each slope induces an interval graph, it easily follows for every G in d-DIR with clique number at most ω that the chromatic number χ (G) of G is at most d ω . We show for every even value of ω how to construct a graph in d-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvořák and Noorizadeh. Furthermore, we show that the χ -binding function of d-DIR is ω ↦ d ω for ω even and ω ↦ d (ω - 1) + 1 for ω odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case d = 2 . [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s00454-025-00737-2 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 13 StartPage: 758 Subjects: – SubjectFull: Intersection graph theory Type: general – SubjectFull: Graph coloring Type: general – SubjectFull: Mathematicians Type: general – SubjectFull: Graph theory Type: general Titles: – TitleFull: The χ-Binding Function of d-Directional Segment Graphs. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Duraj, Lech – PersonEntity: Name: NameFull: Kang, Ross J. – PersonEntity: Name: NameFull: La, Hoang – PersonEntity: Name: NameFull: Narboni, Jonathan – PersonEntity: Name: NameFull: Pokrývka, Filip – PersonEntity: Name: NameFull: Rambaud, Clément – PersonEntity: Name: NameFull: Reinald, Amadeus IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 10 Text: Oct2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 74 – Type: issue Value: 3 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
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