  In the Maker-Breaker vertex colouring game, first publicised by Gardner in
1981, Maker and Breaker alternately colour vertices of a graph using a fixed
palette, maintaining a proper colouring at all times. Maker aims to colour the
whole graph, and Breaker aims to make some vertex impossible to colour. We are
interested in the following question, first asked by Zhu in 1999: if Maker wins
with $k$ colours available, must they also win with $k+1$? This question has
remained open, attracting significant attention and being reposed for many
similar games. While we cannot resolve this problem for the vertex colouring
game, we can answer it in the affirmative for the game of arboricity, resolving
a question of Bartnicki, Grytczuk, and Kierstead from 2008.
  We then consider how one might approach the question of monotonicity for the
vertex colouring game, and work with a related game in which the vertices must
be coloured in a prescribed order. We demonstrate that this `ordered vertex
colouring game' does not have the above monotonicity property, and discuss the
implications of this fact to the unordered game.
  Finally, we provide counterexamples to two open problems concerning a
connected version of the graph colouring game.
