  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.
