TY - JOUR

T1 - The number of F-matchings in almost every tree is a zero residue

AU - Alon, Noga

AU - Haber, Simi

AU - Krivelevich, Michael

PY - 2011

Y1 - 2011

N2 - 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. 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.

AB - 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. 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.

UR - http://www.scopus.com/inward/record.url?scp=79951697723&partnerID=8YFLogxK

U2 - 10.37236/517

DO - 10.37236/517

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AN - SCOPUS:79951697723

SN - 1077-8926

VL - 18

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 1

ER -