# Phase Transitions in the Coloring of Random Graphs

## TL;DR

This paper analyzes the phase transitions in the solution space of graph coloring problems on random graphs, revealing critical thresholds and their implications for computational hardness and algorithms.

## Contribution

It provides a systematic analytical study of the solution space structure and phase transitions in graph coloring, using the cavity method, and discusses algorithmic implications.

## Key findings

- Identification of clustering, condensation, and freezing transitions in solution space.
- Critical connectivities for phase transitions in Erdős-Rényi and regular graphs.
- Freezing transition may be linked to computational hardness.

## Abstract

We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdos-Renyi and regular random graphs and determine their asymptotic values for large number of colors. Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.