2007/04/30 by Lenka Zdeborová, Florent Krzakala, Florent Krząkała · 306 citations
Computer Science · Mathematics · Physics and Astronomy · #Cluster analysis #Combinatorics #Complete coloring #Complex Network Analysis Techniques #Computer science #Discrete mathematics #Entropy (arrow of time) #Finite set #Graph #Graph coloring #Graph power #Greedy coloring #Line graph #Mathematical analysis #Mathematics #Phase transition #Physics #Quantum mechanics #Random graph #Space (punctuation) #Statistical physics #Statistics #Theoretical and Computational Physics #Topological and Geometric Data Analysis #Vertex (graph theory) #cond-mat.dis-nn #cond-mat.stat-mech #cs.CC
paper · pdf · doi:10.1103/physreve.76.031131
published in Physical Review E 76(3), 031131 (American Physical Society) · 36 pages, 15 figures
arxiv created 2007/06/20 · openalex publication_date 2007/09/26 · arxiv updated 2011/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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-Rényi and regular random graphs and determine their asymptotic values for a 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.