On the Holt-Klee Property for Oriented Matroid Programming

TL;DR

This paper extends the Holt-Klee property, originally known for polytopes, to certain oriented matroid programs of rank up to 5, demonstrating the existence of independent monotone paths in their digraphs.

Contribution

It proves the Holt-Klee property for oriented matroid programs of rank up to 5, broadening the class of structures where this property holds.

Findings

Holt-Klee property holds for rank d ≤ 4 in oriented matroid programs.

The result generalizes previous knowledge limited to d ≤ 3 and n ≤ 6.

Provides new insights into the structure of digraphs from oriented matroids.

Abstract

The Holt-Klee theorem says that the graph of a $d$-polytope, with edges oriented by a linear function on $P$ that is not constant on any edge, admits $d$ independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank $d+1$ on $n+2$ elements, which include those from $d$-polytopes with $n$ facets, admit $d$ independent monotone paths from source to sink if $d \le 4$. This was previously only known to hold for $d\le 3$ and $n\le 6$.