从分级词典序看丹齐格图形
On Dantzig figures from graded lexicographic orders
摘要：
本文从$\mathbb{Z}^{d}_{\geq 0}$上的分次字典序序（grlex）和分次逆字典序（grevlex）的初始子集的凸包出发，构造了两类Dantzig图，它们是具有对极顶点对的$(d,2d)$ -多面体.这两个多面体具有相同的顶点数$\mathcal{O}(d^{2})$和相同的边数$\mathcal{O}(d^{3})$，但在组合上不等价。我们提供了一个明确的描述，这两个家庭的顶点和刻面，并描述他们的图形，以及分析他们的基本性质，如半径，直径，存在的哈密顿回路，和色数。此外，我们还分析了这些图的边扩张。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
