  We consider a simple game, the $k$-regular graph game, in which players take
turns adding edges to an initially empty graph subject to the constraint that
the degrees of vertices cannot exceed $k$. We show a sharp topological
threshold for this game: for the case $k=3$ a player can ensure the resulting
graph is planar, while for the case $k=4$, a player can force the appearance of
arbitrarily large clique minors.
