The topology of competitively constructed graphs (Q405221)

Summary: 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.