A Note on the Matching Polytope of a Graph
DOI:
https://doi.org/10.5540/tema.2019.020.01.189Keywords:
Regular graph, Matching polytope, Degree of matchingAbstract
The matching polytope of a graph G, denoted by M(G), is the convex hull of the set of the incidence vectors of the matchings G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G. In this paper, for an arbitrary graph, we prove that the minimum degree of G(M(G)) is equal to the number of edgesof G, generalizing a known result for trees. From this, we identify the vertices of the skeleton with the minimum degree and we prove that the union of stars and triangles characterizes regular skeletons of the matching polytopes of graphs.
References
B. Bollobas, Modern Graph Theory . Graduate Texts in Mathematics, New York: Springer, 1998.
R. Diestel, Graph Theory . New York: Springer-Verlag, 2000.
L. Lovasz and M. D. Plumer, Matching Theory. Ann. Discrete Math. 29 121, Amsterdam: North-Holland, 1986.
B. Grunbaum, Convex Polytopes. New York: Springer-Verlag, 2003.
V. Chvatal, On certain polytopes associated with graphs, Journal of Combinatorial Theory, vol. B 18, pp. 138-154, 1975.
A. Schrijver, Combinatorial optimization: polyhedra and e efficiency .
Algorithms and Combinatorics 24, New York: Springer-Verlag, 2003.
L. Costa, C. M. da Fonseca, and E. A. Martins, The diameter of the acyclic
birkhoff polytope, Linear Algebra Appl, vol. 428, pp. 1524-1537, 2008.
N. Abreu, L. Costa, G. Dahl, and E. Martins, The skeleton of acyclic birkhoff
polytopes, Linear Algebra Appl., vol. 457, pp. 29-48, 2014.
R. Fernandes, Computing the degree of a vertex in the skeleton of acyclic
birkhoff polytopes, Linear Algebra Appl, vol. 475, pp. 119-133, 2015.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.