A Classification of Orientable Regular Embeddings of Complete Multipartite Graphs

TL;DR

This paper provides a comprehensive classification of orientably-regular embeddings of complete multipartite graphs K_{m[n]} for all m ≥ 3 and n ≥ 2, extending previous classifications for specific cases.

Contribution

It offers the first complete classification of orientably-regular embeddings for all complete multipartite graphs K_{m[n]} with m ≥ 3 and n ≥ 2.

Findings

Complete classification of embeddings for K_{m[n]} with m ≥ 3, n ≥ 2

Extension of previous results on bipartite and complete graphs

Unified framework for understanding embeddings of multipartite graphs

Abstract

Let $K_{m[n]}$ be the complete multipartite graph with $m$ parts, while each part contains $n$ vertices. The orientably-regular embeddings of complete graphs $K_{m[1]}$ have been determined by Biggs (1971) \cite{Big1}, James and Jones (1985) \cite{JJ}. During the past twenty years, several papers such as Du et al.(2007, 2010) \cite{DJKNS1,DJKNS2}, Jones et al. (2007, 2008) \cite{JNS1,JNS2}, Kwak and Kwon (2005, 2008) \cite{KK1,KK2} and Nedela et al. (1997, 2002)\cite{NS,NSZ} contributed to the orientably-regular embeddings of complete bipartite graphs $K_{2[n]}$ and the final classification was given by Jones \cite{Jon1} in 2010. Based on our former paper \cite{ZD}, this paper gives a complete classification of orientably-regular embeddings of graphs $K_{m[n]}$ for the general cases $m\ge 3$ and $n\ge 2$.