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
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
Algorithms for Colouring Random k-colourable Graphs
Combinatorics, Probability and Computing
Max k-cut and approximating the chromatic number of random graphs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Solving random satisfiable 3CNF formulas in expected polynomial time
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Why almost all k-colorable graphs are easy
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
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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Feige and Kilian [5] showed that finding reasonable approximative solutions to the coloring problem on graphs is hard. This motivates the quest for algorithms that either solve the problem in most but not all cases, but are of polynomial time complexity, or that give a correct solution on all input graphs while guaranteeing a polynomial running time on average only. An algorithm of the first kind was suggested by Alon and Kahale in [1] for the following type of random k-colorable graphs: Construct a graph $\mathcal{G}_{n,p,k}$ on vertex set V of cardinality n by first partitioning V into k equally sized sets and then adding each edge between these sets with probability p independently from each other. Alon and Kahale showed that graphs from $\mathcal{G}_{n,p,k}$ can be k-colored in polynomial time with high probability as long as p ≥ c/n for some sufficiently large constant c. In this paper, we construct an algorithm with polynomial expected running time for k = 3 on the same type of graphs and for the same range of p. To obtain this result we modify the ideas developed by Alon and Kahale and combine them with techniques from semidefinite programming. The calculations carry over to general k.