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Phase transitions in the coloring of random graphs
Lenka Zdeborová1, Florent Krzakała
1LPTMS, UMR 8626 CNRS et Université Paris-Sud, 91405 Orsay CEDEX, France.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 13, 2007
Summary
Graph coloring transitions reveal how solution spaces change with increasing constraints. The freezing transition, not clustering, likely signals computational difficulty in finding valid colorings.
Area of Science:
- Graph Theory
- Statistical Physics
- Computational Complexity
Background:
- Vertex coloring is a fundamental problem in graph theory.
- Understanding the structure of solution spaces is crucial for complex combinatorial problems.
Purpose of the Study:
- To analytically study the space of proper graph colorings using the cavity method.
- To identify and characterize phase transitions in the solution space of random graphs.
Main Methods:
- Cavity method for analyzing the solution space of proper colorings.
- Investigation of phase transitions as average vertex degree increases.
Main Results:
- Identified clustering, condensation, and freezing transitions in the solution space.
- Computed critical connectivities for random graphs and their asymptotic values.
- Found that the freezing transition, not clustering, may indicate computational hardness.
Conclusions:
- The structure of proper colorings undergoes significant phase transitions with increasing constraints.
- The freezing transition is proposed as a key indicator of computational difficulty for graph coloring algorithms.
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