Half-regular factorizations of the complete bipartite graph

TL;DR

This paper characterizes when a bipartite degree matrix can be realized as a union of half-regular graphs, introduces perturbation methods for transformations, and develops an MCMC algorithm for sampling such graphs.

Contribution

It provides necessary and sufficient conditions for bipartite degree matrices to be realized as unions of half-regular graphs and designs an efficient MCMC sampling method.

Findings

Characterized realizability conditions for bipartite degree matrices.

Developed perturbation techniques for transforming graph realizations.

Designed an MCMC method with polynomially bounded acceptance ratios.

Abstract

We consider a bipartite version of the color degree matrix problem. A bipartite graph $G(U,V,E)$ is half-regular if all vertices in $U$ have the same degree. We give necessary and sufficient conditions for a bipartite degree matrix (also known as demand matrix) to be the color degree matrix of an edge-disjoint union of half-regular graphs. We also give necessary and sufficient perturbations to transform realizations of a half-regular degree matrix into each other. Based on these perturbations, a Markov chain Monte Carlo method is designed in which the inverse of the acceptance ratios are polynomial bounded. Realizations of a half-regular degree matrix are generalizations of Latin squares, and they also appear in applied neuroscience.