Bounded Degree, Triangle Avoidance Graph 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 various games where $${\mathcal{R}}$$ includes bounded degree and triangle avoidance, we determine the winner for all values of n.