IMPROPER COLORING OF WEIGHTED GRID AND HEXAGONAL GRAPHS

Abstract

We study a weighted improper coloring problem motivated by a frequency allocation problem. It consists of associating to each vertex a set of p(v) (weight) distinct colors (frequencies), such that the set of vertices having a given color induces a graph of degree at most k (the case k = 0 corresponds to proper coloring). The objective is to minimize the number of colors. We propose approximation algorithms to compute such a coloring for general graphs. We apply these to obtain good approximation ratio for grid and hexagonal graphs. Furthermore we give exact results for the 2-dimensional grid and the triangular lattice when the weights are all the same.