 
Summary: The number of Fmatchings in almost every tree is a zero residue
Noga Alon
Simi Haber
Michael Krivelevich
Abstract
For graphs F and G an Fmatching in G is a subgraph of G consisting of pairwise vertex disjoint copies
of F. The number of Fmatchings 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, Tn) 0 (mod m), where Tn 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
Fmatchings. This generalizes a recent result of Wagner who showed that the number of independent sets
in a random labeled tree is almost surely a zero residue.
1 Introduction
The number of independent sets in graphs is an important counting parameter. It is particularly wellstudied
for trees and treelike structures. Prodinger and Tichy showed in [10] that the star and the path maximize and
minimize, respectively, the number of independent sets among all trees of a given size. Part of the interest in
this graph invariant stems from the fact that the number of independent sets plays a role in statistical physics as
well as in mathematical chemistry, where it is known as the MerrifieldSimmons index [9]. A problem that arises
in this context is the inverse problem: determine a graph within a given class of graphs (such as the class of all
trees) with a given number of independent sets. It is an open conjecture [7] (see also [6]) that all but finitely
many positive integers can be represented as the number of independent sets of some tree. Recently Wagner
