On strong avoiding games

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.