Col is PSPACE-complete on Triangular Grids

TL;DR

Col is a two-player graph coloring game; the paper proves $PSPACE$-completeness on triangular grid graphs via a reduction from $B2CL$, using a modular tile-based gadget construction where each logical component is realized as an $8\times 8$ gadget tile connected by $WIRE$ gadgets. It relies on planarity from the source problem and introduces techniques to ensure the constructed instance is reachable from an empty board via MAKEUP, REDPLAYS, and EVEN tiles. The main contribution extends hardness of Col to a structured graph family and provides a polynomial-size, grid-aligned embedding, highlighting the robustness of computational hardness for this game. The result constrains expectations for efficient algorithms and informs hardness considerations for grid-based impartial games.

Abstract

We demonstrate that Col is PSPACE-complete on triangular grid graphs via a reduction from Bounded Two-Player Constraint Logic. This is the most structured graph family that Col is known to be computationally hard for.

Figures (16)

• Figure 1: Legend. $a$: empty vertex, $b$: vertex colored blue, $c$: vertex colored red, $d$: empty vertex adjacent to both blue and red; no one can play, $e$: empty vertex adjacent to red; Blue can play, $f$: empty vertex adjacent to blue; Red can play.
• Figure 2: Gadget Template, showing the state of all vertices on the edges of the gadget.
• Figure 3: Gadget Template with neighboring circles in green. Question marks indicate that blue may or may not be able to play at those spots in neighboring gadgets.
• Figure 4: VARIABLE Gadget with the output pair ($o$) labeled. If Blue plays first, they will play on the T, otherwise Red plays there and Blue plays on the F.
• Figure 5: GOAL gadget: In order for Blue to win, they must play at the vertex marked G.

Theorems & Definitions (2)

• Theorem 1: Main
• proof