Connected, Bounded Degree, Triangle Avoidance Games

Abstract

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 a pair of games where $\mathcal{R}$ includes bounded degree, connectedness, and triangle-avoidance, we determine the winner for all values of $n$.