Utilization of permutation classes for solving extremely complex 4-colorable rectangle-free grids

This paper aims at the rectangle-free coloring of grids using four colors. It has been proven that there are bounds for the size of rectangle-free four-colorable grids - outside of these values the grids cannot be colored. For the grids of the size 17 × 17, 17 × 18, 18 × 17, and 18 × 18 it is not yet known whether rectangle-free colorings by four colors exist. A necessary condition for rectangle-free four-colorable grids of the size 18 × 18 is that at least one fourth of the grid elements can be colored by a single color without violating the rectanglefree condition. This simplified task requires to evaluate 2¹⁸*¹⁸ = 2³²⁴ = 3.41758 _* 10⁹⁷ assignments to the grid elements whether at least 18_*18/4 = 81 elements can be colored by the same color without violating the rectangle-free condition. We present in this paper an approach that allows to calculate such extremely large sets of assignments using the restricted memory space of a normal PC and a short period of time. The key to our solution is the utilization of permutation classes. However, even a single permutation class consists of such an extreme number of assignments that it can neither be stored nor enumerated. Hence, it is the main aim of this paper to find a way that allows us to decide whether two grid assignments are members of the same permutation class under these extreme conditions. On the basis of such an approach we are going to answer the question whether an extremely rare four-colorable rectangle-free grid G_18,18 can exist within the huge amount of 4₃₂₄ = 1.16798 _* 10¹⁹⁵ possible four-colorings of such grids.