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{\bf Noga Alon, Simi Haber and Michael Krivelevich}
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{\bf The Number of $f$-Matchings in Almost Every Tree is a Zero Residue}
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For graphs $F$ and $G$ an \emph{$F$-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.
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