Given an increasing graph property $\cal F$, the strong Avoider-Avoider $\cal F$ game is played on the edge set of a complete graph. Two players, Red and Blue, take turns in claiming previously unclaimed edges with Red going first, and the player whose graph possesses $\cal F$ first loses the game. If the property $\cal F$ is "containing a fixed graph $H$", we refer to the game as the $H$ game. We prove that Blue has a winning strategy in two strong Avoider-Avoider games, $P_4$ game and ${\cal CC}_{>3}$ game, where ${\cal CC}_{>3}$ is the property of having at least one connected component on more than three vertices. We also study a variant, the strong CAvoider-CAvoider games, with additional requirement that the graph of each of the players must stay connected throughout the game. We prove that Blue has a winning strategy in the strong CAvoider-CAvoider games $S_3$ and $P_4$, as well as in the $Cycle$ game, where the players aim at avoiding all cycles.
翻译:鉴于图形属性 $\ cal F$, 强大的避免者- 避免者 $\ cal F$ 游戏在完整图表的边缘一组游戏中播放。 两个玩家, 红牌和蓝牌, 以Red为首, 轮流标注先前未宣布的边缘, 而图形拥有 $ cal F$ 的玩家首先会失去游戏。 如果 $\ cal F$ 是“ 包含固定的图形 $H$ ”, 我们将游戏称为 $H 游戏。 我们证明, Blue 在两个强大的避免者- 避免者游戏中, $P_ $_ 4 游戏和$$ CC_ 3} 游戏中, $ CC_ 3$ 是至少一个连接部分的属性。 我们还研究一个变式, 强大的 Caaveer- Caaleser 避免游戏, 额外的要求每个玩家的图表必须在整个游戏中保持连接。 我们证明, Blue 在一个强大的 Caaleser- Caueer 游戏中有一个赢赢的策略 $S_ 3 和 $P_ 4$ 游戏中, $ Cycleclecleclew 中, 。