Generalized ramsey theory XV: Achievement and avoidance games for bipartite graphs

Abstract

Let two opponents, Oh and Ex, play the following game on the complete bipartite graph K_n,n. Oh colors one of the edges green and Ex colors a different edge red, and so on. The goal of each player is to be the first one to construct in his own color a predetermined bipartite graph M with no isolated points. The minimum n for which Oh can win on K_n,n regardless of the moves made by Ex is called the bipartite achievement number of M. In the corresponding misère game, the first player (if any) who forms M is the loser. The minimum n for which Ex can force Oh to make a monochromatic M is called the bipartite avoidance number of M. Bipartite achievement and avoidance numbers of some small graph are presented as well as their bipartite ramsey numbers.