Open Problems About Grid Coloring and The Complexity of Grid Colorings

Not all grids have c-colorings. As an example, for any c, clearly Gc+1,cc+1+1 does not have a c-coloring by two applications of the pigeonhole principle. The following question has been studied [3, 4]. We follow the approach and notation in the paper by Fenner et al [4]. The main question is: For which values of n and m is Gn,m c-colorable? Def 1.3 Let n,m, n′,m′ ∈ N. Gm,n contains Gn′,m′ if n′ ≤ n and m′ ≤ m. Gm,n is contained in Gn′,m′ if n ≤ n′ and m ≤ m′. Proper containment means that at least one of the inequalities is strict.