Proper colouring Painter-Builder game

Abstract: We consider the following two-player game, parametrised by positive integers $n$ and $k$. The game is played between Painter and Builder, alternately taking turns, with Painter moving first. The game starts with the empty graph on $n$ vertices. In each round Painter colours a vertex of her choice by one of the $k$ colours and Builder claims an edge between two previously unconnected vertices. Both players should maintain that during the game the graph admits a proper $k$-colouring. The game ends if either all $n$ vertices have been coloured, or Painter has no legal move. In the former case, Painter wins the game, in the latter one Builder is the winner. We prove that the minimal number of colours $k=k(n)$ allowing Painter's win is of logarithmic order in the number of vertices $n$. Biased versions of the game are also considered.