Combinatorics of flag simplicial 3-polytopes

TL;DR

This paper studies the structure of flag simplicial 3-polytopes, showing they can be reduced to an octahedron via edge contractions, and introduces a partial order to analyze their combinatorial properties.

Contribution

It proves that all flag simplicial 3-polytopes can be reduced to an octahedron through edge contractions and develops a partial order framework for their classification.

Findings

Flag simplicial 3-polytopes can be reduced to octahedron

A partial order on flag simplicial 3-polytopes is introduced

Degree estimates for vertices in the Hasse graph are provided

Abstract

In the focus of this paper is the operation of edge contraction. One can show that simplicial 3-polytope is flag iff contraction of any its edge gives simplicial 3-polytope. Our main result states that any flag simplicial 3-polytope can be reduced to octahedron by sequence of edge contractions. Using this operation we introduce a partial order on the set of flag simplicial 3-polytopes and study Hasse graph of corresponding poset. We estimate input and output degrees of vertices of this Hasse graph.