On Dantzig figures from graded lexicographic orders

Abstract

We construct two families of Dantzig figures, which are $(d,2d)$-polytopes with an antipodal vertex pair, from convex hulls of initial subsets for the graded lexicographic (grlex) and graded reverse lexicographic (grevlex) orders on $\mathbb{Z}^{d}_{\geq 0}$. These two polytopes have the same number of vertices, $\mathcal{O}(d^{2})$, and the same number of edges, $\mathcal{O}(d^{3})$, but are not combinatorially equivalent. We provide an explicit description of the vertices and the facets for both families and describe their graphs along with analyzing their basic properties such as the radius, diameter, existence of Hamiltonian circuits, and chromatic number. Moreover, we also analyze the edge expansions of these graphs.