Maker-Breaker games on random geometric graphs

In a Maker-Breaker game on a graph G, Breaker and Maker alternately claim edges of
G. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired
property. We consider four Maker-Breaker games played on random geometric graphs. For each of
our four games we show that if we add edges between n points chosen uniformly at random in the
unit square by order of increasing edge-length then, with probability tending to one as n→∞, the
graph becomes Maker-win the very moment it satisfies a simple necessary condition. In particular,
with high probability, Maker wins the connectivity game as soon as the minimum degree is at least
two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker
wins the perfect matching game as soon as the minimum degree is at least two and every edge has at
least three neighbouring vertices; and Maker wins the H-game as soon as there is a subgraph from a
finite list of “minimal graphs.” These results also allow us to give precise expressions for the limiting
probability that G(n, r) is Maker-win in each case, where G(n, r) is the graph on n points chosen
uniformly at random on the unit square with an edge between two points if and only if their distance
is at most r.