An Improved Bound on the Sizes of Matchings Guaranteeing a Rainbow Matching

Clemens, Dennis; Ehrenmüller, Julia

doi:10.37236/5080

Cited by 16 publications

(13 citation statements)

References 11 publications

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“…Theorem 2 is a strengthening of an earlier result by the first two authors [3], which proves the above statement when the multigraph G is bipartite (and thus each clique consists of two vertices). The latter was motivated by famous conjectures of Ryser [10] and of Brualdi and Stein [2,11] on Latin squares and by the following conjecture of Aharoni and Berger [1].…”

Section: Introductionsupporting

confidence: 67%

On sets not belonging to algebras and rainbow matchings in graphs

Clemens

Ehrenmüller

Pokrovskiy

2017

Journal of Combinatorial Theory, Series B

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Abstract. Motivated by a question of Grinblat, we study the minimal number v(n) that satisfies the following. If A 1 , . . . , An are equivalence relations on a set X such that for every i ∈ [n] there are at least v(n) elements whose equivalence classes with respect to A i are nontrivial, then A 1 , . . . , An contain a rainbow matching, i.e. there exist 2n distinct elements x 1 , y 1 , . . . , xn, yn ∈ X with x i ∼ A i y i for each i ∈ [n]. Grinblat asked whether v(n) = 3n − 2 for every n ≥ 4. The best-known upper bound was v(n) ≤ 16n/5 + O(1) due to Nivash and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that v(n) = 3n + o(n).

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Section: Introductionsupporting

confidence: 67%

On sets not belonging to algebras and rainbow matchings in graphs

Clemens

Ehrenmüller

Pokrovskiy

2017

Journal of Combinatorial Theory, Series B

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“…We merge Conjecture 1 and Theorem 1 into a unified context and ask the following. (We note that this question was also raised independently in [6].) Question 1.…”

mentioning

confidence: 83%

Rainbow matchings in bipartite multigraphs

2016

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“…Aharoni, Charbit, and Howard [2] proved that matchings of size 7n/4 are sufficient to guarantee a rainbow matching of size n. Kotlar and Ziv [11] improved this to 5n/3. Clemens and Ehrenmüller [7] further improved the constant to 3n/2 which is currently the best known bound. Theorem 1.7 (Clemens and Ehrenmüller, [7]).…”

Section: Introductionmentioning

confidence: 99%

“…Clemens and Ehrenmüller [7] further improved the constant to 3n/2 which is currently the best known bound. Theorem 1.7 (Clemens and Ehrenmüller, [7]). Let G be a bipartite graph consisting of n matchings, each with at least 3n/2 + o(n) edges.…”

Section: Introductionmentioning

confidence: 99%

Rainbow Matchings and Rainbow Connectedness

Pokrovskiy

2017

Electron. J. Combin.

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Aharoni and Berger conjectured that every bipartite graph which is the union of n matchings of size n + 1 contains a rainbow matching of size n. This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. In the case when the matchings are much larger than n + 1, the best bound is currently due to Clemens and Ehrenmüller who proved the conjecture when the matchings are of size at least 3n/2+o(n). When the matchings are all edge-disjoint and perfect, then the best result follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least n + o(n).In this paper we show that the conjecture is true when the matchings have size n + o(n) and are all edge-disjoint (but not necessarily perfect). We also give an alternative argument to prove the conjecture when the matchings have size at least φn + o(n) where φ ≈ 1.618 is the Golden Ratio.Our proofs involve studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

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An Improved Bound on the Sizes of Matchings Guaranteeing a Rainbow Matching

Cited by 16 publications

References 11 publications

On sets not belonging to algebras and rainbow matchings in graphs

On sets not belonging to algebras and rainbow matchings in graphs

Rainbow matchings in bipartite multigraphs

Rainbow Matchings and Rainbow Connectedness

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