Bichromatic Triangle Games

We study a combinatorial game called Bichromatic Triangle Game, defined as follows. Two players R and B construct a triangulation on a given planar point set V. Starting from no edges, players R and B take turns drawing one edge that connects two points in V. Player R uses color red and player B uses color blue. The first player who completes one empty monochromatic triangle is the winner. We show that either player can force a tie in the Bichromatic Triangle Game when the points of V are in convex position and also in the case when there is exactly one inner point in the set V. As an easy consequence of those results, we obtain that the outcome of the Bichromatic Complete Triangulation Game (a version of the Bichromatic Triangle Game in which players draw edges until they complete a triangulation) is also a tie for the same two cases regarding set V.