Bounds on spectrum graph coloring

Abstract

We propose two vertex-coloring problems for graphs, endorsing the spectrum of colors with a matrix of interferences between pairs of colors. In the Threshold Spectrum Coloring problem, the number of colors is fixed and the aim is to minimize the maximum interference at a vertex (interference threshold). In the Chromatic Spectrum Coloring problem, a threshold is settled and the aim is to minimize the number of colors (among the available ones) for which respecting that threshold is possible. We prove general upper bounds for the solutions to each problem, valid for any graph and any matrix of interferences. We also show that both problems are NP-hard and perform experimental results, proposing a DSATUR-based heuristic for each problem, in order to study the gap between the theoretical upper bounds and the values obtained.