Fractional Perfect b-Matching Polytopes. I: General Theory

TL;DR

This paper develops a comprehensive theoretical framework for fractional perfect b-matching polytopes in graphs, characterizing their structure, vertices, and face lattices through spanning subgraphs and combinatorial conditions.

Contribution

It introduces general theorems for nonemptiness, dimension, and vertex characterization of fractional perfect b-matching polytopes, expanding understanding of their combinatorial and geometric properties.

Findings

Vertices correspond to subgraphs with components that are either acyclic or contain exactly one odd cycle.

Provides formulas for the dimension and face lattice structure of the polytopes.

Establishes conditions for nonemptiness of the fractional perfect b-matching polytope.

Abstract

The fractional perfect b-matching polytope of an undirected graph G is the polytope of all assignments of nonnegative real numbers to the edges of G such that the sum of the numbers over all edges incident to any vertex v is a prescribed nonnegative number b_v. General theorems which provide conditions for nonemptiness, give a formula for the dimension, and characterize the vertices, edges and face lattices of such polytopes are obtained. Many of these results are expressed in terms of certain spanning subgraphs of G which are associated with subsets or elements of the polytope. For example, it is shown that an element u of the fractional perfect b-matching polytope of G is a vertex of the polytope if and only if each component of the graph of u either is acyclic or else contains exactly one cycle with that cycle having odd length, where the graph of u is defined to be the spanning subgraph of G whose edges are those at which u is positive.