The Complexity of Grid Coloring

A c-coloring of the grid GN,M = [N] × [M] is a mapping of GN,M into [c] such that no four corners forming a rectangle have the same color. In 2009 a challenge was proposed to find a 4-coloring of G17,17. Though a coloring was produced, finding it proved to be difficult. This raises the question of whether there is some complexity lower bound. Consider the following problem: given a partial c-coloring of the GN,M grid, can it be extended to a full c-coloring? We show that this problem is NP-complete. We also give a Fixed Parameter Tractable algorithm for this problem with parameter c.