A Proof of the Strict Monotone 5-step Conjecture

TL;DR

This paper uses computer enumeration to prove that the maximum strictly monotone diameter for 5-dimensional polytopes with 10 facets is 5, confirming the strict monotone 5-step conjecture.

Contribution

It provides the first computer-assisted proof of the strict monotone 5-step conjecture and extends enumeration techniques to related diameter bounds.

Findings

Maximum strictly monotone diameter for (5,10) is 5

Enumeration confirms diameter bounds for (4,9) and (5,10)

Shortens the proof of the strict monotone 4-step conjecture

Abstract

A computer search through the oriented matroid programs with dimension 5 and 10 facets shows that the maximum strictly monotone diameter is 5. Thus $\Delta_{sm}(5,10)=5$. This enumeration is analogous to that of Bremner and Schewe for the non-monotone diameter of 6-polytopes with 12 facets. Similar enumerations show that $\Delta_{sm}(4,9)=5$ and $\Delta_m(4,9)=\Delta_m(5,10)=6.$ We shorten the known non-computer proof of the strict monotone 4-step conjecture.