Why almost all k-colorable graphs are easy to color

Amin Coja-Oghlan, Michael Krivelevich, Dan Vilenchik*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)523-565
Number of pages43
JournalTheory of Computing Systems
Issue number3
StatePublished - Apr 2010


FundersFunder number
USA-Israel BSF2002-133, 526/05
Israel Science Foundation


    • Average case analysis
    • Random graphs
    • Spectral analysis
    • k-colorability


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