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arXiv · 1011.6554

The number of maximum matchings in a tree

Clemens Heuberger·Stephan Wagner
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Abstract

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order $n$ is at most $O(1.391664^n)$ (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by Górska and Skupień on the number of maximal matchings (maximal with respect to set inclusion).

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Clemens Heuberger, Stephan Wagner. 2010-11-30. The number of maximum matchings in a tree. https://doi.org/10.1016/j.disc.2011.07.028

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