An NP-Complete Problem in Grid Coloring

A c-coloring of 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 via the internet to find a 4-coloring of G17,17. This attracted considerable attention from the popular mathematics community. A coloring was produced; however, finding it proved to be difficult. The question arises: is the problem of grid coloring is difficult in general? We present three results that support this conjecture: (1) Given a partial ccoloring of an Gn,m grid, can it be extended to a full c-coloring? We show this problem is NP-complete. (2) The statement Gn,m is c-colorable can be expressed as a Boolean formula with nmc variables. We show that if the Gn,m is not c-colorable then any tree resolution proof of the corresponding formula is of size 2Ω(c). (We then generalize this result for other monochromatic shapes.) (3) We show that any tree-like cutting planes proof that c+ 1 by c (c+1 2 ) + 1 is not c-colorable must be of size 2Ω(c / log c). Note that items (2) and (3) yield statements from Ramsey Theory which are of size polynomial in their parameters and require exponential size in various proof systems.