On improper colouring of unit disk graphs

Motivated by a satellite communications problem, we consider a generalised colouring problem on unit disk graphs. A colouring is k-improper if no vertex receives the same colour as k+1 of its neighbours. The k-improper chromatic number χ(G) is the least number of colours needed in a k-improper colouring of a graph G. The main subject of this work is analysing the complexity of computing χ for the class of unit disk graph and some related classes, e.g. hexagonal graphs and interval graphs. We show NP-completeness in many restricted cases and also provide both positive and negative approximability results. Due to the challenging nature of this topic, many seemingly simple questions remain: for example, it remains open to determine the complexity of computing χ for unit interval graphs. Key-words: improper colouring, defective colouring, unit disk graph, interval graph, triangular lattice, hexagonal graph, weighted colouring. This work was partially supported by Région Provence-Alpes-Côte D’Azur. ∗ mascotte, i3s (cnrs-unsa) – inria, 2004 Route des Lucioles, 06902 Sophia Antipolis Cedex, France. E-mail: fhavet@sophia.inria.fr. † Department of Statistics, 1 South Parks Road, Oxford OX1 3TG, United Kingdom. E-mail: kang@stats.ox.ac.uk. This author is partially supported by NSERC of Canada and the Commonwealth Scolarship Commission (UK). ‡ Institute for Theoretical Computer Science (iti) and Department of Applied Mathematics (kam), Faculty of Mathematics and Physics, Malostranské náměst́ı 25, 118 00 Prague, Czech Republic. E-mail: sereni@kam.mff.cuni.cz. This author is supported by the European project ist fet Aeolus. Coloration impropre des graphes d’intersection de disques unitaires Résumé : Nous modélisons un problème de télécommunications à l’aide des graphes d’intersection de disques unitaires, et d’une généralisation de la notion de coloration. Une coloration est kimpropre si chaque sommet a la couleur d’au plus k de ses voisins. Le nombre chromatique k-impropre χ(G) du graphe G est le plus petit nombre de couleurs pour lequel G admette une coloration k-impropre. Nous étudions la complexité de la détermination de χ lorsque l’on se restreint à la classe des graphes d’intersection de disques unitaires, ainsi qu’à d’autres classes de graphes utiles pour la modélisation des problèmes de télécommunications (comme par exemple les graphes hexagonaux). Nous prouvons que ce problème est NP-complet dans de nombreux cas, et fournissons également des résultats positifs et négatifs d’approximation. Mots-clés : coloration impropre, graphe d’intersection de disques, graphe d’intervalles, réseau triangulaire, graphe hexagonal, coloration pondérée. Improper colouring of unit disk graphs 3

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