Theory of Computing Systems

, Volume 46, Issue 3, pp 523–565 | Cite as

Why Almost All k-Colorable Graphs Are Easy to Color

  • Amin Coja-Oghlan
  • Michael Krivelevich
  • Dan VilenchikEmail author


Coloring a k-colorable graph using k colors (k≥3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly cn edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs lie in a single “cluster”, and agree on all but a small, though constant, portion of the vertices. We also describe a polynomial time algorithm that whp finds a proper k-coloring of such a random k-colorable graph, thus asserting that most such graphs are easy to color. This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics.


Average case analysis k-colorability Random graphs Spectral analysis 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Michael Krivelevich
    • 2
  • Dan Vilenchik
    • 3
    Email author
  1. 1.School of InformaticsUniversity of EdinburghEdinburghUK
  2. 2.School of Mathematical Sciences, Sackler Faculty of Exact SciencesTel-Aviv UniversityTel-AvivIsrael
  3. 3.Department of MathematicsUCLALos AngelesUSA

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