Revisiting Mayer: Symmetric solutions for sporadic cases of the Map Color Theorem

Abstract

The original proof of the genus of the complete graphs $K_n$ depended on Mayer's \emph{ad hoc} solutions for $n = 18, 20, 23$. Recently, an improved solution for $K_{20}$ was found by the author. The purpose of this note is to use the theory of current graphs to interpret the aforementioned result and to provide new embeddings of $K_{18}$ and $K_{23}$.