The Online Clique Avoidance Game on Random Graphs
Consider the following one player game on an empty graph with n vertices. The edges are presented one by one to the player in a random order. One of two colors, red or blue, has to be assigned to each edge immediately. The player’s object is to color as many edges as possible without creating a monochromatic clique K ℓ of some fixed size ℓ. We prove a threshold phenomenon for the expected duration of this game. We show that there is a function N 0 = N 0(ℓ, n) such that the player can asymptotically almost surely survive up to N(n) ≪ N 0 edges by playing greedily and that this is best possible, i.e., there is no strategy such that the game would last for N(n) ≫ N 0 edges.
KeywordsRandom Graph Graph Property Edge Coloring Base Graph Greedy Strategy
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- 12.Friedgut, E., Rödl, V., Ruciński, A., Tetali, P.: A sharp threshold for random graphs with monochromatic triangle in every edge coloring. Memoirs of the AMS (to appear)Google Scholar