An NP-Complete Problem in Grid Coloring

TL;DR

This paper proves that extending a partial grid coloring to a full proper coloring is an NP-complete problem, highlighting the computational difficulty of grid coloring challenges.

Contribution

It establishes the NP-completeness of the problem of extending partial grid colorings to full proper colorings.

Findings

Proves the NP-completeness of extending partial grid colorings.

Highlights the computational difficulty of grid coloring problems.

Addresses a longstanding challenge related to 4-colorings of grids.

Abstract

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 show that the problem of, given a partial coloring of a grid, can it be extended to a full (proper) coloring, is NP-complete.