Quadrangular embeddings of complete graphs and the Even Map Color Theorem (with details)

TL;DR

This paper determines the minimal genus surfaces for complete graphs with quadrangular and even face embeddings, extending previous results and providing new examples that support the Even Map Color Theorem.

Contribution

It extends known quadrangular embedding results to all complete graphs, determines minimal genus for even face embeddings, and offers new minimal quadrangulations via lexicographic products.

Findings

Complete graphs $K_n$ have minimal genus embeddings with all faces of degree at least 4.

The paper provides sharpness examples for Hutchinson's bound on chromatic number.

New minimal quadrangulations are constructed using lexicographic products of graphs.

Abstract

Hartsfield and Ringel constructed orientable quadrangular embeddings of the complete graph $K_n$ for $n\equiv 5 \pmod 8$, and nonorientable ones for $n \ge 9$ and $n\equiv 1 \pmod 4$. These provide minimal quadrangulations of their underlying surfaces. We extend these results to determine, for every complete graph $K_n$, $n \ge 4$, the minimum genus, both orientable and nonorientable, for the surface in which $K_n$ has an embedding with all faces of degree at least $4$, and also for the surface in which $K_n$ has an embedding with all faces of even degree. These last embeddings provide sharpness examples for a result of Hutchinson bounding the chromatic number of graphs embedded with all faces of even degree, completing the proof of the Even Map Color Theorem. We also show that if a connected simple graph $G$ has a perfect matching and a cycle then the lexicographic product $G[K_4]$ has orientable and nonorientable quadrangular embeddings; this provides new examples of minimal quadrangulations.