Minimizing Unsatisfaction in Colourful Neighbourhoods

Abstract

Colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance. Here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods, given a predetermined number of colours. In the analytical framework of a tree approximation, carried out at both zero and finite temperatures, solutions obtained by population dynamics give rise to estimates of the threshold connectivity for the incomplete to complete transition, which are consistent with those of existing algorithms. The nature of the transition as well as the validity of the tree approximation are investigated.

28 pages, 12 figures, substantially revised with additional explanation

References in corpus (4)

• Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems
• Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
• Potts Glass on Random Graphs
• Inference and Optimization of Real Edges on Sparse Graphs - A Statistical Physics Perspective