Ramsey properties of random graphs
Journal of Combinatorial Theory Series B
On-line coloring of sparse random graphs and random trees
Journal of Algorithms
Zero knowledge and the chromatic number
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
On the hardness of approximating minimization problems
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Improved hardness results for approximating the chromatic number
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Ramsey Games Against a One-Armed Bandit
Combinatorics, Probability and Computing
The online clique avoidance game on random graphs
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Upper bounds for online ramsey games in random graphs
Combinatorics, Probability and Computing
Online ramsey games in random graphs
Combinatorics, Probability and Computing
Coloring random graphs online without creating monochromatic subgraphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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Consider the following generalized notion of graph colorings: a vertex coloring of graph G is valid w.r.t. some fixed nonempty graph F if no color class induces a copy of F in G, i.e., there is no monochromatic copy of F in G. We propose and analyze an algorithm for computing valid colorings of a random graph Gn, p on n vertices with edge probability p in an online fashion. For a large family of graphs F including cliques and cycles of arbitrary size, the proposed algorithm is optimal in the following sense: for any integer r ≥ 1, there is a constant β = β(F, r) such that the algorithm a.a.s. (asymptotically almost surely) computes a valid r-coloring of Gn, p w.r.t. F online if p ≪ n-β, and any online algorithm will a.a.s. fail to do so if p ≫ n-β. That is, we observe a threshold phenomenon determined by the function n-β.