Graph Theoretical Models of Closed n-Dimensional Manifolds: Digital Models of a Moebius Strip, a Torus, a Projective Plane a Klein Bottle and n-Dimensional Spheres

TL;DR

This paper develops graph theoretical models of n-dimensional manifolds that preserve topological properties, providing digital representations of complex surfaces like spheres, tori, and Klein bottles with minimal point counts.

Contribution

It introduces a method to construct digital models of continuous n-dimensional manifolds using intersection graphs of LCL collections, maintaining their topological features.

Findings

Digital n-sphere has at least 2n+2 points

Digital projective plane has at least eleven points

Digital Klein bottle has at least sixteen points

Abstract

In this paper, we show how to construct graph theoretical models of n-dimensional continuous objects and manifolds. These models retain topological properties of their continuous counterparts. An LCL collection of n-cells in Euclidean space is introduced and investigated. If an LCL collection of n-cells is a cover of a continuous n-dimensional manifold then the intersection graph of this cover is a digital closed n-dimensional manifold with the same topology as its continuous counterpart. As an example, we prove that the digital model of a continuous n-dimensional sphere is a digital n-sphere with at least 2n+2 points, the digital model of a continuous projective plane is a digital projective plane with at least eleven points, the digital model of a continuous Klein bottle is the digital Klein bottle with at least sixteen points, the digital model of a continuous torus is the digital torus with at least sixteen points and the digital model of a continuous Moebius band is the digital Moebius band with at least twelve points.