A refinement of the edge theorem for order and chain polytopes

Source abstract

For a finite poset, we partition the edges of the order polytope and of the chain polytope into classes indexed by the connected convex subsets of the poset, and we establish that corresponding classes have the same cardinality. This yields a closed formula for the common number of edges, and it identifies the bijection of Hibi, Li, Sahara and Shikama, given by them through an explicit formula, as a disjoint union of simple bijections between corresponding classes, which explains why it is a bijection and how it acts on edge directions and lengths. As consequences, the two polytopes have equally many edge directions, with matching multiplicities; the chain polytope has at least as many edges parallel to each coordinate subspace as the order polytope; and the edge lengths of the chain polytope are dominated by those of the order polytope. Strict inequality occurs in the last two comparisons exactly when the poset contains a three-element chain. We discuss implications for linear and convex combinatorial optimization over ideals and antichains. We also express the number of edges in terms of the comparability graph, in a form that extends to stable-set polytopes of arbitrary graphs, give a recursion for series-parallel posets and closed formulas for layered, zigzag and crown posets, characterize the distributive lattices for which the two polytopes are unimodularly equivalent, and demonstrate that, although the two polytopes have the same number of edges, either one can have the larger diameter, by an arbitrary amount; on the other hand, both diameters are bounded by the width of the poset, and they coincide for ordinal sums, series-parallel posets, and zigzag and crown posets.