The χ-Binding Function of d-Directional Segment Graphs.
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| Title: | The χ-Binding Function of d-Directional Segment Graphs. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-025-00737-2 |
