Planarity, Colorability, And Minor Games

摘要

Let m and 6 be positive integers, and let F be a hypergraph. In an (m, b) Maker-Breaker game F two players, called Maker and Breaker, take turns selecting previously unclaimed vertices of F. Maker selects m vertices per move, and Breaker selects b vertices per move. The game ends when every vertex has been claimed by one of the players. Maker wins if he claims all of the vertices of some hyperedge of F; otherwise Breaker wins. An (m, b) Avoider-Enforcer game F is played in a similar way. The only difference is in the determination of the winner: Avoider loses if he claims all of the vertices of some hyperedge of F; otherwise Enforcer loses. In this paper we consider the Maker-Breaker and Avoider-Enforcer versions of the planarity game, the k-colorability game, and the K_t-minor game.