Irregular triangulations of complete graphs on 12s vertices in orientable surfaces

TL;DR

This paper introduces a simpler method for creating triangulations of complete graphs on 12s vertices in orientable surfaces using index 1 abelian current graphs, improving upon previous complex techniques.

Contribution

The authors develop a new, simplified construction for genus embeddings of K_{12s} graphs using abelian current graphs, avoiding large index or nonabelian groups.

Findings

Constructed triangulations for K_{12s} with s ≥ 4

Simplified the process compared to previous methods

Achieved embeddings with index 1 abelian current graphs

Abstract

We present a family of index 1 abelian current graphs whose derived embeddings can be modified into triangulations of $K_{12s}$ for $s \geq 4$. Our construction is significantly simpler than previous methods for finding genus embeddings of $K_{12s}$, which utilized either large index or nonabelian groups.