# Octahedralizing 3-colorable 3-polytopes

## TL;DR

This paper proves that every 3-colorable 3-polytope can be subdivided into a cross-polytopal complex without adding new boundary vertices, advancing understanding of polytope subdivisions.

## Contribution

It establishes that all 3-colorable 3-polytopes can be octahedralized without boundary vertex addition, a new result in polytope subdivision theory.

## Key findings

- Positive answer for dimension 3
- Octahedralization without boundary vertices
- Advances in polytope subdivision understanding

## Abstract

We investigate the question of whether any $d$-colorable simplicial $d$-polytope can be octahedralized, i.e., it can be subdivided to a $d$-dimensional geometric cross-polytopal complex. We give a positive answer in dimension $3$, with the additional property that the octahedralization introduces no new vertices on the boundary of the polytope.