Discrete Mathematics and Combinatorics
While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and then investigate the differences between forests with game chromatic number 3 and 4. In doing so, we present a minimal example of a forest with game chromatic number 4, criteria for determining in polynomial time the game chromatic number of a forest without vertices of degree 3, and an example of a forest with maximum degree 3 and game chromatic number 4. This gives partial progress on the open question of the computational complexity of the game chromatic number of a forest.
The game chromatic number of trees and forests
Discrete Mathematics and Theoretical Computer Science, 2015, volume 17, issue 2, pages 31-48
Dunn, Charles; Larsen, Victor; Retter, Troy; Lindke, Kira; and Toci, Dustin, "The Game Chromatic Number of Trees and Forests" (2015). Faculty Publications. Published Version. Submission 7.