The Number of $f$-Matchings in Almost Every Tree is a Zero Residue

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, 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 F -matchings. 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.

On the Number of $F$-Matchings in a Tree

“…Theorem 1 answers the questions of Alon, Haber and Krivelevich [1] on the number of F -matchings and on induced F -matchings. Note also that by restricting ourselves to trees with no disjoint copies of F in the proof, we avoid multiplication in Z m (and hence number theoretic difficulties).…”

“…We prove Theorem 1 and some natural extensions in Section 2. In Section 3 we generalize the results in [15] and [1] on the asymptotic behaviour of s r (F, T ). While zero is the dominating residue, we show that the other residue classes still grow exponentially in Section 4.…”

“…Searching for an intuitive explanation of Wagner's result, Alon, Haber and Krivelevich [1] considered F -matchings which generalize both independent sets and matchings. Let F, G be graphs.…”

“…Since both versions are multiplicative over components, if there is a component whose number of F -matchings is a multiple of m, then the residue of the whole graph is necessarily 0. Developing this property, a nullifying tree was constructed in [1,Lemma 8] such that if contained in T , then the number of F -matchings in T is a zero residue. Then it was proved a.a.e.…”

“…The authors of [1] raised the question: Given k, m > 0 and a tree F , is there a tree T whose number of F -matchings is congruent to k (mod m)? The analogous question for induced F -matchings was also raised.…”