Genus embeddings of complete graphs minus a matching

TL;DR

This paper constructs genus embeddings of complete graphs minus a matching for specific n, using orientable triangular embeddings of octahedral graphs and handle augmentations, also relating to surface crossing numbers.

Contribution

It introduces a method to produce genus embeddings of complete graphs minus a matching for all n divisible by 6 and at least 18, expanding understanding of graph embeddings.

Findings

Existence of orientable triangular embeddings for specified n

Construction of genus embeddings via handle augmentation

Determination of surface crossing numbers for these graphs

Abstract

We show that for all $n \equiv 0 \pmod{6}$, $n \geq 18$, there is an orientable triangular embedding of the octahedral graph on $n$ vertices that can be augmented with handles to produce a genus embedding of the complete graph of the same order. For these values of $n$, the intermediate embeddings of the construction also determine some surface crossing numbers of the complete graph on $n$ vertices and the genus of all graphs on $n$ vertices and minimum degree $n-2$.