Game matching number of graphs

We study a competitive optimization version of $\alpha'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted \Max(G) and \Min(G), respectively. We show that always |\Max(G)-\Min(G)|\le 1. We obtain a sufficient condition for \Max(G)=\alpha'(G) that is preserved under cartesian product. In general, \Max(G)\ge \frac23\alpha'(G), with equality for many split graphs, while \Max(G)\ge\frac34\alpha'(G) when $G$ is a forest. Whenever $G$ is a 3-regular $n$-vertex connected graph, \Max(G) \ge n/3, and there are such examples with \Max(G)\le 7n/18. For an $n$-vertex path or cycle, the answer is roughly $n/7$.Comment: 16 pages; this version improves explanations at a number of points and adds a few more example