The numbers of edges of the order polytope and the chain poyltope of a finite partially ordered set

TL;DR

This paper proves that the order and chain polytopes of a finite poset have the same number of edges and characterizes when their 1-skeletons have identical degree sequences, linking combinatorial and geometric properties.

Contribution

It establishes the equality of the number of edges for order and chain polytopes and characterizes their 1-skeleton degree sequence equivalence.

Findings

Number of edges of ${ m O}(P)$ equals that of ${ m C}(P)$

Degree sequences of 1-skeletons are equal iff the polytopes are unimodularly equivalent

Provides a geometric-combinatorial characterization of polytope equivalence

Abstract

Let $P$ be an arbitrary finite partially ordered set. It will be proved that the number of edges of the order polytope ${\mathcal O}(P)$ is equal to that of the chain polytope ${\mathcal C}(P)$. Furthermore, it will be shown that the degree sequence of the finite simple graph which is the $1$-skeleton of ${\mathcal O}(P)$ is equal to that of ${\mathcal C}(P)$ if and only if ${\mathcal O}(P)$ and ${\mathcal C}(P)$ are unimodularly equivalent.