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An NP-Complete Problem in Grid Coloring
A c-coloring of G(n,m)=n x m is a mapping of G(n,m) into {1,...,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 G(17,17). 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) an NP completeness result, (2) a lower bound
on Tree-resolution, (3) a lower bound on Tree-CP proofs. 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.Comment: 25 page
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