The Graph Coloring Game on 4 x n-Grids
Caroline Brosse, Nicolas Almeida Martins, Nicolas Nisse, Rudini Sampaio · Procedia Computer Science · 2025
The graph coloring game is a famous two-player game (re)introduced by Bodlaender in 1991. Given a graph G and k ϵ N , Alice and Bob alternately (starting with Alice) color an uncolored vertex with some color in {1, • • •, 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 ϵ N, deciding whether χ g (G) ≤ k is PSPACE-complete. Surprisingly, this parameter is not well understood even in “simple” graph classes. Let P n denote the path with n ≥ 1 vertices. For instance, in the case of Cartesian grids, it is easy to show that χ g ( P m □ P n ) ≤ 5 since χ g (G) ≤ ∆ + 1 for any graph G with maximum degree ∆. However, the exact value is only known for small values of m , namely χ g (P 1 □ P n ) = 3, χ g (P 2 □ P n ) = 4 and χ g ( P 3 □ Pn ) = 4 for n ≥ 4 [Raspaud, Wu, 2009]. Here, we prove that, for every n ≥ 18, χ g ( P 4 □ P n ) = 4.