The Number of $f$-Matchings in Almost Every Tree is a Zero Residue
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Published:2011-02-14
Issue:1
Volume:18
Page:
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ISSN:1077-8926
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Container-title:The Electronic Journal of Combinatorics
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language:
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Short-container-title:Electron. J. Combin.
Author:
Alon Noga,Haber Simi,Krivelevich Michael
Abstract
For graphs $F$ and $G$ an $F$-matching in $G$ is a subgraph of $G$ consisting of pairwise vertex disjoint copies of $F$. The number of $F$-matchings in $G$ is denoted by $s(F,G)$. We show that for every fixed positive integer $m$ and every fixed tree $F$, the probability that $s(F,\mathcal{T}_n) \equiv 0 \pmod{m}$, where $\mathcal{T}_n$ is a random labeled tree with $n$ vertices, tends to one exponentially fast as $n$ grows to infinity. A similar result is proven for induced $F$-matchings. As a very special special case this implies that the number of independent sets in a random labeled tree is almost surely a zero residue. A recent result of Wagner shows that this is the case for random unlabeled trees as well.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
2 articles.
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