Minimal quadrangulations of surfaces

TL;DR

This paper determines the minimal number of vertices needed for quadrangulations of all surfaces, providing explicit formulas and using a novel diagonal technique for proofs.

Contribution

It establishes formulas for the minimal order of quadrangulations on all surfaces, including orientable and nonorientable, using a new proof technique.

Findings

Exact values for minimal quadrangulation orders on key surfaces

A general formula for all surfaces based on Euler characteristic

Introduction of the diagonal technique for proofs

Abstract

A quadrangular embedding of a graph in a surface $\Sigma$, also known as a quadrangulation of $\Sigma$, is a cellular embedding in which every face is bounded by a $4$-cycle. A quadrangulation of $\Sigma$ is minimal if there is no quadrangular embedding of a (simple) graph of smaller order in $\Sigma$. In this paper we determine $n(\Sigma)$, the order of a minimal quadrangulation of a surface $\Sigma$, for all surfaces, both orientable and nonorientable. Letting $S_0$ denote the sphere and $N_2$ the Klein bottle, we prove that $n(S_0)=4, n(N_2)=6$, and $n(\Sigma)=\lceil (5+\sqrt{25-16\chi(\Sigma)})/2\rceil$ for all other surfaces $\Sigma$, where $\chi(\Sigma)$ is the Euler characteristic. Our proofs use a `diagonal technique', introduced by Hartsfield in 1994. We explain the general features of this method.