Improper coloring of unit disk graphs

Abstract

Abstract Motivated by a satellite communications problem, we consider a generalized coloring problem on unit disk graphs. A coloring is k ‐improper if no more than k neighbors of every vertex have the same colour as that assigned to the vertex. The k ‐improper chromatic number χ k ( G ) is the least number of colors needed in a k ‐improper coloring of a graph G . The main subject of this work is analyzing the complexity of computing χ k for the class of unit disk graphs 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. Because of the challenging nature of this topic, many seemingly simple questions remain: for example, it remains open to determine the complexity of computing χ k for unit interval graphs. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009