The connected greedy coloring game

Abstract

In 2020, Costa et al. answered a long-standing open question, proposed by Bodlaender in 1991: the graph coloring game is PSPACE-complete. They also proved that the greedy coloring game, introduced by Havet and Zhu in 2013, is also PSPACE-complete. In 2019, Andres and Lock proposed five variants of the graph coloring game, which were proved PSPACE-complete by Lima et al. in 2022. In this paper, we extend these variants of the graph coloring game to the greedy coloring game and prove that all of them are PSPACE-complete, even if the number of colors is the chromatic number. Moreover, based on the connected variant of the graph coloring game proposed by Charpentier et al. in 2020, we introduce the connected greedy coloring game and prove that two natural variants of this game are PSPACE-complete. Despite this, we prove that in split graphs these two variants of the connected greedy coloring game are polynomial time solvable and we characterize the split graphs for which the associated connected game Grundy numbers are the minimum possible value (the chromatic number). Finally, we also characterize the split graphs which are bad, a notion introduced by Le and Trotignon in 2018 regarding connected greedy colorings.