Variations on Ringel's earth-moon problem (Q1969794)

Given a pair of surfaces \(\Sigma_1\) and \(\Sigma_2\), an empire map consists of a map on each of them, with a given 1-1 correspondence between their countries. A coloring of an empire map is a coloring of each of these maps such that corresponding countries on \(\Sigma_1\) and \(\Sigma_2\) are colored the same. The minimum number of colors needed to color all empire maps is denoted \(\chi(\Sigma_1,\Sigma_2)\), and abbreviated to \(\chi_2(\Sigma)\) when \(\Sigma_1=\Sigma_2=\Sigma\). Ringel's earth-moon problem asks for \(\chi_2(S^2)\), which is known only to lie between 9 and 12. Using maps constructed with the aid of current graphs, the authors show that \(\chi_2(T^2)=13\) and \(\chi_2(P^2)=12\) (where \(T^2\) denotes the 2--torus and \(P^2\) the 2--dimensional projective plane). They also prove a Heawood--type bound for \(\chi_2(\Sigma)\) for any surface \(\Sigma\). Finally, some results are obtained about generalizations of this notion to more than two surfaces.