TY - JOUR

T1 - Multicolored matchings in hypergraphs

AU - Alon, Noga

N1 - Publisher Copyright:
© 2011, Mathematical Sciences Publishers. All rights reserved.

PY - 2011

Y1 - 2011

N2 - For a collection of (not necessarily distinct) matchings M = (M1, M2,:::, Mq) in a hypergraph, where each matching is of size t, a matching M of size t contained in the union [qi=1Mi is called a rainbow matching if there is an injective mapping from M to M assigning to each edge e of M a matching Mi 2 M containing e. Let f(r, t) denote the maximal k for which there exists a collection of k matchings, each of size t, in some r-partite r-uniform hypergraph, such that there is no rainbow matching of size t. Aharoni and Berger showed that f(r, t) 2r–1 (t – 1), proved that the equality holds for r = 2 as well as for t = 2 and conjectured that the equality holds for all r, t. We show that in fact f(r, t) is much bigger for most values of r and t, establish an upper bound and point out a relation between the problem of estimating f(r, t) and several results in additive number theory, which provides new insights into some of those results.

AB - For a collection of (not necessarily distinct) matchings M = (M1, M2,:::, Mq) in a hypergraph, where each matching is of size t, a matching M of size t contained in the union [qi=1Mi is called a rainbow matching if there is an injective mapping from M to M assigning to each edge e of M a matching Mi 2 M containing e. Let f(r, t) denote the maximal k for which there exists a collection of k matchings, each of size t, in some r-partite r-uniform hypergraph, such that there is no rainbow matching of size t. Aharoni and Berger showed that f(r, t) 2r–1 (t – 1), proved that the equality holds for r = 2 as well as for t = 2 and conjectured that the equality holds for all r, t. We show that in fact f(r, t) is much bigger for most values of r and t, establish an upper bound and point out a relation between the problem of estimating f(r, t) and several results in additive number theory, which provides new insights into some of those results.

KW - Additive Latin transversals

KW - Rainbow matchings

UR - http://www.scopus.com/inward/record.url?scp=84893541128&partnerID=8YFLogxK

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AN - SCOPUS:84893541128

SN - 2220-5438

VL - 1

SP - 3

EP - 10

JO - Moscow Journal of Combinatorics and Number Theory

JF - Moscow Journal of Combinatorics and Number Theory

IS - 1

ER -