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Why Almost All k-Colorable Graphs Are Easy

  • Amin Coja-Oghlan
  • Michael Krivelevich
  • Dan Vilenchik
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)

Abstract

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 are clustered in one cluster, and agree on all but a small, though constant, number of vertices. We also describe a polynomial time algorithm that finds a proper k-coloring for (1 − o(1))-fraction of such random k-colorable graphs, thus asserting that most of them are “easy”. 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. One explanation for this phenomena, backed up by partially non-rigorous analytical tools from statistical physics, is the complicated clustering of the solution space at that regime, unlike the more “regular” structure that denser graphs possess. Thus in some sense, our result rigorously supports this explanation.

Keywords

Exchange Rate Random Graph Polynomial Time Algorithm Chromatic Number Graph Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Michael Krivelevich
    • 2
  • Dan Vilenchik
    • 3
  1. 1.Institute for Informatics, Humboldt-University, BerlinGermany
  2. 2.School Of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-AvivIsrael
  3. 3.School of Computer Science, Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-AvivIsrael

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