The number of F-matchings in almost every tree is a zero residue

TL;DR

This paper proves that for almost all large trees, the count of F-matchings (including induced matchings) is divisible by any fixed integer m with high probability, extending previous work on independent sets.

Contribution

It generalizes Wagner's result by showing the divisibility property holds for all fixed trees and F-matchings, not just independent sets.

Findings

Probability that s(F,T_n) = 0 mod m tends to one exponentially fast

Results hold for both F-matchings and induced F-matchings

Extends previous work on independent sets in random trees

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,T_n) = 0 mod m, where 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. 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.