Title: 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,
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.