A Markov chain on the solution space of edge-colorings of bipartite graphs

TL;DR

This paper introduces a Markov chain on edge colorings of bipartite graphs, analyzing its properties and implications for sampling solutions, with applications to Latin rectangles.

Contribution

It establishes the irreducibility and linear diameter growth of the Markov chain, and provides polynomial bounds on the Metropolis-Hastings acceptance ratio.

Findings

Markov chain on edge colorings is irreducible.

Diameter of the chain grows linearly with edges.

Polynomial bounds on Metropolis-Hastings acceptance ratio.

Abstract

In this paper, we exhibit an irreducible Markov chain $M$ on the edge $k$-colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis-Hastings algorithm when the algorithm is applied on $M$ with the uniform distribution of all possible edge $k$-colorings of $G$. A special case of our results is the solution space of the possible completions of Latin rectangles.