  The problems of the genus of the complete graphs and minimum triangulations
for each surface were both solved using the theory of current graphs, and each
of them divided into twelve different cases, depending on the residue modulo 12
of the number of vertices. Cases 8 and 11 were of particular difficulty for
both problems, with multiple families of current graphs developed to solve
these cases. We solve these cases in a unified manner with families of current
graphs applicable to both problems. Additionally, we give new constructions to
both problems for Cases 6 and 9, which greatly simplify previous constructions
by Ringel, Youngs, Guy, and Jungerman. All these new constructions are index 3
current graphs sharing nearly all of the structure of the simple solution for
Case 5 of the Map Color Theorem.
