Simultaneous current graph constructions for minimum triangulations and complete graph embeddings

Abstract

The problems of calculating the genus of the complete graphs and of finding a minimum triangulation 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 addition to Cases 6 and 9, in a unified manner, greatly simplifying 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.