Impartial coloring games

TL;DR

This paper explores six impartial coloring game variants, analyzing their outcome classes and computational complexity, with some focus on the Grundy function, expanding understanding of these combinatorial games.

Contribution

Introduces five new impartial coloring game rulesets derived from existing schemes and studies their outcome classes and complexity.

Findings

Outcome classes characterized for special cases

Computational complexity analyzed for each ruleset

Grundy function examined in certain cases

Abstract

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, and sequential coloring. For each, we study the outcome classes for special cases and general computational complexity. In some cases we pay special attention to the Grundy function.