Impartial coloring games

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.

💡 Research Summary

The paper introduces a unified framework for impartial coloring games, defining six rule sets that stem from classic graph‑coloring constraints: proper coloring, oriented coloring, 2‑distance coloring, weak coloring, sequential coloring, and a newly proposed limited‑color‑usage variant. In each game two players alternately select an uncolored vertex and assign it a permissible color; the move set is identical for both players, making the games impartial in the Sprague‑Grundy sense.

For each rule the authors conduct a three‑fold analysis. First, they determine outcome classes on elementary graph families such as paths, cycles, trees, and complete graphs. For proper coloring with at least three colors, paths are always first‑player wins, while cycles are winning for the first player only when the length is odd. Weak coloring on trees admits a linear‑time algorithm based on a bottom‑up Grundy computation. Second, they derive explicit Grundy functions or recursive formulas. The sequential variant decomposes naturally into independent intervals whose Grundy numbers are combined by XOR, while 2‑distance coloring reduces to proper coloring on the square of the graph, allowing the authors to bound Grundy values by the maximum degree of the squared graph. Oriented coloring’s Grundy numbers collapse to 0 or 1 depending on the presence of directed cycles within strong components. The limited‑color‑usage game exhibits a parameter‑dependent Grundy structure: when the total allowed uses of each color are constant, a dynamic‑programming solution runs in O(n·L) time; otherwise the problem becomes PSPACE‑complete.

Third, the paper establishes computational complexity results. Proper, oriented, and 2‑distance impartial coloring are shown PSPACE‑complete via reductions from Geography and Node Kayles, confirming that the impartial versions inherit the hardness of their partisan counterparts. Weak coloring is NP‑hard on general graphs but polynomial on trees. Sequential coloring is polynomial because the imposed vertex order transforms the game into a disjoint sum of subgames. The limited‑color‑usage variant is PSPACE‑complete when the usage bound is part of the input, yet fixed‑parameter tractable for small bounds.

Experimental data on all graphs up to 20 vertices illustrate the distribution of Grundy numbers and the frequency of first‑player wins across the six rule sets, highlighting cases where theoretical hardness does not manifest in practice. The authors conclude with several open directions: average‑case analysis on random graph models, parameterized complexity with respect to color‑budget, and extensions that combine impartial coloring with color‑switching dynamics. Overall, the work bridges combinatorial game theory and graph coloring, expanding the landscape of impartial games and providing a comprehensive taxonomy of their algorithmic and strategic properties.