The online clique avoidance game on random graphs

  • Authors:

  • Martin Marciniszyn;Reto Spöhel;Angelika Steger

  • Affiliations:

  • Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland;Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland;Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland

  • Venue:
  • 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
  • Year:
  • 2005

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Abstract

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 Kℓ of some fixed size ℓ. We prove a threshold phenomenon for the expected duration of this game. We show that there is a function N0 = N0(ℓ, n) such that the player can asymptotically almost surely survive up to N(n) ≪ N0 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) ≫ N0 edges.