On Dantzig figures from graded lexicographic orders

TL;DR

This paper constructs two new families of Dantzig figures from convex hulls of initial subsets under graded lexicographic orders, analyzing their properties and differences.

Contribution

It introduces explicit constructions of Dantzig figures from graded lexicographic orders and compares their combinatorial and graph-theoretic properties.

Findings

Both polytopes have O(d^2) vertices and O(d^3) edges.

The two polytopes are not combinatorially equivalent.

Analysis of graph properties including radius, diameter, Hamiltonian circuits, chromatic number, and edge expansion.

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.