Extending triangulations of the 2-sphere to the 3-disk preserving a 4-coloring

TL;DR

This paper proves that any 4-colored triangulation of a 2-sphere can be extended to a 3-disk triangulation with the same coloring, bridging 2D and 3D topological structures.

Contribution

It introduces a method to extend 2D sphere triangulations with 4-colorings into 3D disks while maintaining the coloring scheme.

Findings

Any 4-colored sphere triangulation can be realized as the boundary of a 3-disk triangulation.

The extension preserves the vertex coloring scheme.

The result applies to all triangulations of the 2-sphere with a strict 4-coloring.

Abstract

In this paper we prove that any triangulation of a 2-dimensional sphere with a strict 4-coloring on its vertices can seen as the boundary of a triangulation of a 3-dimensional disk with the same vertices and preserving the 4-coloring.