Polytopes of Magic Labelings of Graphs and the Faces of the Birkhoff Polytope

Abstract

In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. This enables us to generate and enumerate perfect matchings of a graph via magic labelings of the graph. We explore the correspondence of magic labelings of graphs with magic squares and define polytopes of magic labelings to give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.

Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors

“…Carathéodory's theorem ensures that every n×n doubly stochastic matrix can be written as a convex combination of at most n 2 − 2n + 2 permutation matrices. Geometrical and combinatorial properties of the Birkhoff polytope have been extensively studied; see, e.g., [2,6,7,8,10,11,20,17]; see also [18, pp. 47-52] for a brief account.…”

“…In enumerating vertices of the polytope L n , different approaches have been undertaken: (1). Combinatorial method via Latin squares (see, e.g., [3, Theorem 0.1] or [1, Theorem 2.0.10]); (2). Analytic approach by using hyperplane and induction [9]; (3).…”

On Volumes of Permutation Polytopes

“…. , 1 (n) } of the all-ones vector 1 in R h−1 , together with n copies {−e (1) Now consider the chamber in the Gale dual whose extreme rays are {−e 1 , −e 2 , . .…”

“…Proposition 4.3. The Gale dual of P (G) is the vector configuration consisting of n copies {1 (1) , 1 (2) , . .…”

“…It is defined as the convex hull of all n×n permutation matrices, or equivalently, as the convex hull of the natural permutation representation of the symmetric group S n ; see [4,6,7,9,10,16,18,27,31] and references therein for a summary of its known properties. Subpolytopes of B n have been shown to have remarkably beautiful properties; see [1,6,8,10,11,19,26,28,36] and references therein. This is particularly true for permutation polytopes, those polytopes that arise by taking convex hulls of permutation representations of special subgroups of S n with concrete sets of generators.…”