Simultaneous current graph constructions for minimum triangulations and complete graph embeddings

TL;DR

This paper introduces unified current graph constructions that solve complex cases in the problems of genus of complete graphs and minimum triangulations, simplifying previous methods and providing new solutions.

Contribution

It presents a unified approach to difficult cases in graph genus and triangulation problems using index 3 current graphs, simplifying and extending prior work.

Findings

Unified solutions for Cases 8 and 11 in both problems.

New simplified constructions for Cases 6 and 9.

Current graphs share structure with the Map Color Theorem solution.

Abstract

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.