The Graph Coloring Game on $4\times n$-Grids

TL;DR

This work resolves the four-by-n grid case for the graph coloring game by proving χ_g(P_4 × P_n) ≤ 4 for all n and exactly χ_g(P_4 × P_n) = 4 for all n ≥ 18, using a novel block-based strategy. Central to the method is a detailed combinatorial framework of blocks, borders, and local configurations that preserves a robust invariant under Bob’s moves, enabling Alice to color the entire grid with four colors. The approach introduces concepts such as safe, sound, and sick vertices and a taxonomy of border configurations (alpha, beta, gamma, delta, pi) plus seven particular configurations to handle edge scenarios, culminating in a long, case-heavy but systematic proof of correctness. The results illuminate the structure of adversarial coloring on structured grids and point to interesting avenues for general m,n and related graph classes, with several open questions guiding future work.

Abstract

The graph coloring game is a famous two-player game (re)introduced by Bodlaender in $1991$. Given a graph $G$ and $k \in \mathbb{N}$, Alice and Bob alternately (starting with Alice) color an uncolored vertex with some color in $\{1,\cdots,k\}$ such that no two adjacent vertices receive a same color. If eventually all vertices are colored, then Alice wins and Bob wins otherwise. The game chromatic number $χ_g(G)$ is the smallest integer $k$ such that Alice has a winning strategy with $k$ colors in $G$. It has been recently (2020) shown that, given a graph $G$ and $k\in \mathbb{N}$, deciding whether $χ_g(G)\leq k$ is PSPACE-complete. Surprisingly, this parameter is not well understood even in ``simple" graph classes. Let $P_n$ denote the path with $n\geq 1$ vertices. For instance, in the case of Cartesian grids, it is easy to show that $χ_g(P_m \times P_n) \leq 5$ since $χ_g(G)\leq Δ+1$ for any graph $G$ with maximum degree $Δ$. However, the exact value is only known for small values of $m$, namely $χ_g(P_1\times P_n)=3$, $χ_g(P_2\times P_n)=4$ and $χ_g(P_3\times P_n) =4$ for $n\geq 4$ [Raspaud, Wu, 2009]. Here, we prove that, for every $n\geq 18$, $χ_g(P_4\times P_n) =4$.

Figures (4)

• Figure 1: Illustration of the notions of safeness and soundness in the middle of a block. Note that a vertex is depicted by a square. The bold lines figure the surrounding of the block. A $0$ in a cell means that the corresponding vertex is not colored, $c$ and $c'$ denote any colors (not $0$). It can be deduced from the figure that $c'\neq c$ since there are two adjacent vertices colored $c$ and $c'$ respectively (and the coloring is proper). A white cell indicates that the status of the corresponding vertex is not constrained (it may be colored or not). The $j$ above denotes the index of the corresponding column.
• Figure 2: Illustration of the different configurations in the case of a right border (column $j$). The colors $c,c',c"\ne 0$ are pairwise distinct. Each configuration has its symmetrical counterpart (according to the horizontal symmetry axis of the grid). The configurations for the left borders are defined symmetrically (according to the vertical symmetry axis of the grid).
• Figure 3: Configurations $\Delta$, $\Delta'$, $\Delta'_2$, $\Lambda$ and $\Lambda'$, where $c'\ne c$ and $c"\ne c'$. The vertex marked "safe" in configurations $\Delta'$ and $\Delta'_2$ is not a doctor of some vertex in column $j-2$ (indeed, for it being safe, its neighbor in column $j-2$ must be colored). Moreover, the vertex marked "safe" in configuration $\Lambda'$ guarantees that $v_{2,j+1}$ is sound (with doctors $v_{1,j+1}$ and $v_{3,j+1}$). Configurations $\Lambda, \Lambda', \Lambda_2, \Lambda'_2$ and $\Delta$ have no symmetrical counter-part according to the vertical axis. Recall that column $j-3$ cannot be empty in configurations $\Lambda_2$ and $\Lambda'_2$. The red vertices are called the sick vertices.
• Figure 4: Example of a sequence of moves in the game from (a) to (d), where Alice's (resp. Bob's) moves are represented in red (resp. blue).

Theorems & Definitions (8)

• Claim 1
• proof
• Claim 2
• proof
• Lemma 1
• proof
• Theorem 1
• proof