Circular Coloring of Random Graphs: Statistical Physics Investigation

TL;DR

This paper investigates circular coloring of random graphs using statistical physics methods, revealing phase transitions, symmetry breaking, and the complexity of solution spaces, with implications for understanding coloring problems and optimization algorithms.

Contribution

It introduces a novel analysis of circular coloring on random graphs, identifying phase transitions and the limitations of one-step replica symmetry breaking.

Findings

Spontaneous symmetry breaking occurs at low temperatures and many colors.

5-circular coloring of 3-regular graphs is colorable but requires complex solution space analysis.

Simulated annealing efficiently finds solutions despite complex solution space structure.

Abstract

Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.