On the vertex degrees of the skeleton of the matching polytope of a graph

TL;DR

This paper investigates the structure of the skeleton of the matching polytope of a graph, providing formulas for vertex degrees, identifying minimum degree vertices, and characterizing regular skeletons.

Contribution

It introduces a formula to compute vertex degrees in the skeleton of the matching polytope for any graph and characterizes the regular skeletons.

Findings

Minimum degree of the skeleton equals the number of edges in the graph.

Vertices with minimum degree are characterized.

Regular skeletons are fully characterized.

Abstract

The convex hull of the set of the incidence vectors of the matchings of a graph G is the matching polytope of the graph, M(G). The graph whose vertices and edges are the vertices and edges of M(G) is the skeleton of the matching polytope of G, denoted G(M(G)). Since the number of vertices of G(M(G)) is huge, the structural properties of these graphs have been studied in particular classes. In this paper, for an arbitrary graph G, we obtain a formulae to compute the degree of a vertex of G(M(G)) and prove that the minimum degree of G(M(G)) is equal to the number of edges of G. Also, we identify the vertices of the skeleton with the minimum degree and characterize regular skeletons of the matching polytopes.