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

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.