Channels, Billiards, and Perfect Matching 2-Divisibility

TL;DR

This paper introduces combinatorial tools called channels to analyze the parity and divisibility of perfect matchings in graphs, connecting graph theory with billiard dynamics and providing new bounds and algorithms.

Contribution

It presents a novel combinatorial proof relating channels to 2-divisibility of perfect matchings, and establishes a surprising link between graph matchings and billiard paths.

Findings

Channels provide a lower bound on 2-divisibility of perfect matchings.

A connection between billiard paths and graph matchings is established.

An efficient algorithm for counting channels in grid regions is developed.

Abstract

Let $m_G$ denote the number of perfect matchings of the graph $G$. We introduce a number of combinatorial tools for determining the parity of $m_G$ and giving a lower bound on the power of 2 dividing $m_G$. In particular, we introduce certain vertex sets called channels, which correspond to elements in the kernel of the adjacency matrix of $G$ modulo $2$. A result of Lov\'asz states that the existence of a nontrivial channel is equivalent to $m_G$ being even. We give a new combinatorial proof of this result and strengthen it by showing that the number of channels gives a lower bound on the power of $2$ dividing $m_G$ when $G$ is planar. We describe a number of local graph operations which preserve the number of channels. We also establish a surprising connection between 2-divisibility of $m_G$ and dynamical systems by showing an equivalency between channels and billiard paths. We exploit this relationship to show that $2^{\frac{\gcd(m+1,n+1)-1}{2}}$ divides the number of domino tilings of the $m\times n$ rectangle. We also use billiard paths to give a fast algorithm for counting channels (and hence determining the parity of the number of domino tilings) in simply connected regions of the square grid.