How Long Can a Graph be Kept Planar?

Abstract

The graph (non-)planarity game is played on the complete graph $K_n$ between an Enforcer and an Avoider, each of whom take one edge per round. The game ends when the edges chosen by Avoider form a non-planar subgraph. We show that Avoider can play for $3n-26$ turns, improving the previous bound of $3n-28\sqrt n$.