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Title: Channels, Billiards, and Perfect Matching 2-Divisibility
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 Lovász 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.  more » « less
Award ID(s):
1700302
PAR ID:
10277356
Author(s) / Creator(s):
;
Date Published:
Journal Name:
The Electronic Journal of Combinatorics
Volume:
28
Issue:
2
ISSN:
1077-8926
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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