Year: 2025
Type: article
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 $${\mathbb R}^2$$ 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 $$\omega $$ ω that the chromatic number $$\chi (G)$$ χ ( G ) of G is at most $$d\omega $$ d ω . We show for every even value of $$\omega $$ 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 $$\chi $$ χ -binding function of d-DIR is $$\omega \mapsto d\omega $$ ω ↦ d ω for $$\omega $$ ω even and $$\omega \mapsto d(\omega -1)+1$$ ω ↦ d ( ω - 1 ) + 1 for $$\omega $$ ω odd. This extends an earlier result by Kostochka and Nešetřil, which treated the special case $$d=2$$ d = 2 .
Institutions: Jagiellonian University, University of Amsterdam, Masaryk University, Centre National de la Recherche Scientifique, Institut national de recherche en sciences et technologies du numérique, Université Côte d'Azur, Laboratoire d'Informatique, Signaux et Systèmes de Sophia Antipolis, COATI: Combinatoire, Optimisation et Algorithmes pour les Télécommunications, Université de Montpellier, Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4410460792