On the minimum order of a quadrangulation on a given closed 2-manifold

TL;DR

This paper presents a partial formula to determine the minimum number of vertices needed for quadrangulations on closed orientable 2-manifolds of a given genus, extending previous work by Hartsfield and Ringel.

Contribution

It introduces an extended partial formula for calculating the minimal vertices in quadrangulations, advancing the understanding of graph embeddings on surfaces.

Findings

Provides a new partial formula for minimum vertices in quadrangulations.

Extends previous results by Hartsfield and Ringel.

Enhances methods for analyzing graph embeddings on 2-manifolds.

Abstract

A partial formula is provided to calculate the smallest number of vertices possible in a quadrangulation on the closed orientable 2-manifold of given genus. This extends the previously known partial formula due to N. Hartsfield and G. Ringel [J. Comb. Theory, Ser. B, 1989, 46, 84-95].