#### Date of Original Version

5-2-2014

#### Type

Article

#### Rights Management

This is the accepted version of the article which has been published in final form at http://dx.doi.org/10.1002/rsa.20572

#### Abstract or Description

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*.

#### DOI

10.1002/rsa.20572

#### Published In

Random Structures and Algorithms, 45, 4, 553-607.