A note on monotonicity in Maker-Breaker graph colouring games

Abstract: 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.