Planar graph coloring with an uncooperative partner

We show that the game chromatic number of a planar graph is at most 33. More generally, there exists a function f : ℕ → ℕ so that for each n ∈ ℕ, if a graph does not contain a homeomorph of K n , then its game chromatic number is at most f ( n ). In particular, the game chromatic number of a graph is bounded in terms of its genus. Our proof is motivated by the concept of p ‐arrangeability, which was first introduced by Guantao and Schelp in a Ramsey theoretic setting.