# Vertex-Facet Assignments For Polytopes

## TL;DR

This paper investigates the existence of vertex-facet matchings in polytopes, providing conditions for their existence and constructing counterexamples in higher dimensions, advancing understanding of polytope structure.

## Contribution

It establishes general conditions for vertex-facet assignments and demonstrates their existence in simple, simplicial, and low-dimensional polytopes, with counterexamples in higher dimensions.

## Key findings

- Vertex-facet assignments exist for all simple and simplicial polytopes.
- Such assignments exist for all polytopes of dimension up to 6.
- Counterexamples are constructed for all dimensions 7 and above.

## Abstract

Motivated by the search for reduced polytopes, we consider the following question: For which polytopes exists a vertex-facet assignment, that is, a matching between vertices and non-incident facets, so that the matching covers either all vertices, or all facets? We provide general conditions for the existence of such an assignment. We conclude that such exist for all simple and simplicial polytopes, as well as all polytopes of dimension $d\le 6$. We construct counterexample in all dimensions $d\ge 7$.