Path achievement games (Q2712505)

Consider the following game, played with two players. The game starts with the empty graph on \(n\) vertices, and the players alternatingly add an edge until the graph has a path of \(p\) edges. This paper analyses which player has a winning strategy for \(p\leq 6\) and gives a winning strategy. For \(p=1,2,3\), this is trivial. For \(p=4\), the paper shows that player 1 has a winning strategy if \(n\bmod 4= 2\) or \(n\) mod \(4=3\). For \(p=5\), player 1 wins iff \(n\bmod 4 \neq 1\), i.e., for \(p=4\), \(p=5\), the game has periodicity 4. For \(p=6\) a similar result is given, but now with periodicity 84.