Normality of $k$-Matching Polytopes of Bipartite Graphs
Combinatorics
2023-06-22 v1
Abstract
The -matching polytope of a graph is the convex hull of all its matchings of a given size when they are considered as indicator vectors. In this paper, we prove that the -matching polytope of a bipartite graph is normal, that is, every integer point in its -dilate is the sum of integers points of the original polytope. This generalizes the known fact that Birkhoff polytopes are normal. As a preliminary result, we prove that for bipartite graphs the -matching polytope is equal to the fractional -matching polytope, having thus the -representation of the polytope. This generalizes the Birkhoff-Von Neumann Theorem which establish that every doubly stochastic matrix can be written as a convex combination of permutation matrices.
Keywords
Cite
@article{arxiv.2306.11910,
title = {Normality of $k$-Matching Polytopes of Bipartite Graphs},
author = {Juan Camilo Torres},
journal= {arXiv preprint arXiv:2306.11910},
year = {2023}
}