On strong avoiding games
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Given an increasing graph property F, the strong Avoider-Avoider 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 F first loses the game. If the property 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 CC_>3 game, where 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.
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