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Random graph coloring: statistical physics approach.

J van Mourik1, D Saad

  • 1The Neural Computing Research Group, Aston University, Birmingham B4 7ET, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 7, 2003
PubMed
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This study explores random graph vertex coloring using statistical physics, comparing analytical findings with simulations. It provides exact solutions for two-coloring and approximations for general graph coloring problems.

Area of Science:

  • Graph theory
  • Statistical physics
  • Computational complexity

Background:

  • Vertex coloring is a fundamental problem in graph theory with applications in resource allocation and scheduling.
  • Understanding the behavior of graph coloring in random graph ensembles is crucial for theoretical computer science.

Purpose of the Study:

  • To investigate the vertex coloring problem in random graphs.
  • To compare analytical results from statistical physics with simulation and enumeration methods.
  • To develop exact and approximate solutions for graph coloring.

Main Methods:

  • Statistical physics techniques
  • Probability theory
  • Exact enumeration
  • Monte Carlo simulations

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Main Results:

  • An exact analytical expression for the two-coloring problem in random graphs.
  • General replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and K-body edges.
  • Comparative analysis of different methodologies for solving graph coloring problems.

Conclusions:

  • The study validates the application of statistical physics methods to random graph coloring.
  • It highlights the strengths and limitations of various computational approaches.
  • Provides new analytical tools for understanding complex graph coloring scenarios.