Bounded Degree, Triangle Avoidance Graph Games

We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game begins with two players and an empty graph on n vertices. The two players take turns choosing edges within K n , building up a simple graph. The edges must be chosen according to a set of restrictions \({\mathcal{R}}\) . The winner is the last player to choose an edge that does not violate any of the restrictions in \({\mathcal{R}}\) . For fixed n and \({\mathcal{R}}\) , one of the players has a winning strategy. For various games where \({\mathcal{R}}\) includes bounded degree and triangle avoidance, we determine the winner for all values of n.