Let $G$ be a graph with three disjoint triangles(i.e. the graph is not connectd and has three connected components each of which is a triangle). If each vertex of G is assigned a red or a green color, then we say that an edge is colored if its ends have different colors. Person A and Person B color the vertices of G in the following manner. A proposes a color (red or green) and B chooses the vertex to apply this color. After 9 turns, all the vertices of G are colored and the number of colored edges is counted. Suppose A would like to maximize the number of colored edges while B would like to minimize the number of colored edges. Assuming optimal play from both players, how many edges will be colored?
This is a puzzle I would like to know the answer to. I suspect that finally two triangles will have the same color for all vertices while the third one will have two same and one different color.
What is the correct logic?