Hardness of some variants of the graph coloring game

TL;DR

This paper establishes that several variants of the graph coloring game, including the greedy and connected versions, are PSPACE-complete, confirming their computational hardness even with constraints like the chromatic number.

Contribution

It proves PSPACE-completeness for five variants of the graph coloring game and a connected version, resolving open questions about their computational complexity.

Findings

Variants are PSPACE-complete even with the chromatic number.

Connected version of the game is PSPACE-complete.

Confirms computational hardness of multiple graph coloring game variants.

Abstract

Very recently, a long-standing open question proposed by Bodlaender in 1991 was answered: the graph coloring game is PSPACE-complete. In 2019, Andres and Lock proposed five variants of the graph coloring game and left open the question of PSPACE-hardness related to them. In this paper, we prove that these variants are PSPACE-complete for the graph coloring game and also for the greedy coloring game, even if the number of colors is the chromatic number. Finally, we also prove that a connected version of the graph coloring game, proposed by Charpentier et al. in 2019, is PSPACE-complete.