Almost all k-colorable graphs are easy to color
Journal of Algorithms
The solution of some random NP-hard problems in polynomial expected time
Journal of Algorithms
Coloring random and semi-random k-colorable graphs
Journal of Algorithms
Random Structures & Algorithms
A Spectral Technique for Coloring Random 3-Colorable Graphs
SIAM Journal on Computing
Zero knowledge and the chromatic number
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
A sharp threshold for k-colorability
Random Structures & Algorithms
Almost all graphs with average degree 4 are 3-colorable
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Heuristics for semirandom graph problems
Journal of Computer and System Sciences
A spectral technique for random satisfiable 3CNF formulas
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Coloring sparse random k-colorable graphs in polynomial expected time
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Complete convergence of message passing algorithms for some satisfiability problems
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
On the random satisfiable process
Combinatorics, Probability and Computing
A spectral method for MAX2SAT in the planted solution model
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
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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.