This is an extended version of this question, more carefully presented for professional mathematicians.
Consider a two-player game on the complete undirected graph $K_5$, in which edges are initially uncolored. Player A chooses an uncolored edge and colors it blue, then player B chooses an uncolored edge and colors it red. Players alternate until one player (the loser) is forced to have colored three edges forming a triangle with the same color. (Such a game can end in a draw, as $K_5$ can be expressed as the union of two 5-cycles, as Gerhard Paseman points out.) Assume that on each move a player avoids completing such a triangle, if possible.
Questions
- How many distinguishable games are there? Two games are distinguishable if there is no permutation of the vertex labels (and hence edge labels) that yields the same sequence of moves.
- What is the discrete distribution of total moves in distinguishable games, assuming each valid game is equally likely?
- One can use Ramsey theory to count the number of end configurations that end in a draw. What is the number of distinguishable games than end in a loss for player A, loss for player B, or draw?
- Generalize the above to $K_n$ for arbitrary $n \in \mathbb{Z}^+$.