Large matchings in bipartite graphs have a rainbow matching

Kotlar, Daniel; Ran, Ziv

doi:10.1016/j.ejc.2013.11.011

Cited by 20 publications

(14 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We start with assuming for a contradiction that a maximum rainbow matching in the given graph G = (A ∪ B, E) is of size n − 1. A rainbow matching of this size is known to exist [9]. We fix such a matching R and find two sequence e 1 , .…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

See 1 more Smart Citation

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

Clemens

Ehrenmüller

2016

Electron. J. Combin.

View full text Add to dashboard Cite

A conjecture by Aharoni and Berger states that every family of n matchings of size n + 1 in a bipartite multigraph contains a rainbow matching of size n. In this paper we prove that matching sizes of 3 2 + o(1) n suffice to guarantee such a rainbow matching, which is asymptotically the same bound as the best known one in case we only aim to find a rainbow matching of size n − 1. This improves previous results

show abstract

Section: Proof Of Theorem 13mentioning

confidence: 99%

“…For a contradiction, let us assume that there is no such matching. As shown in [9], there must exist a rainbow matching R of size n − 1. We may assume without loss of generality that none of the edges of F 0 appears in R. Let t be the smallest positive integer with 1/(2t − 1) ≤ ε.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

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

Clemens

Ehrenmüller

2016

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…Indeed, if the largest rainbow matching M contains less than n − 1 edges, then there would be an edge with an unused color that is not adjacent to any of the edges in M and the edge could be added to M to get a larger rainbow matching. After subsequent effort by several authors (see, e.g., [6], [10], [14], [18], [22], and [24]), Pokrovskiy proved an asymptotic version of Conjecture 1.2 in [23], where he proved that the conclusion of the conjecture holds if there are at least n + o(n) edges of each color.…”

Section: Introduction 1state Of the Artmentioning

confidence: 99%

Rainbow matchings in edge-colored simple graphs

Chakraborti¹,

Loh²

2020

Preprint

View full text Add to dashboard Cite

There has been much research on the topic of finding a large rainbow matching (with no two edges having the same color) in a properly edge-colored graph, where a proper edge coloring is a coloring of the edge set such that no same-colored edges are incident. Barát, Gyárfás, and Sárközy conjectured that in every proper edge coloring of a multigraph (with parallel edges allowed, but not loops) with 2q colors where each color appears at least q times, there is always a rainbow matching of size q. Recently, Aharoni, Berger, Chudnovsky, Howard, and Seymour proved a relaxation of the conjecture with 3q − 2 colors. Our main result proves that 2q + o(q) colors are enough if the graph is simple, confirming the conjecture asymptotically for simple graphs. This question restricted to simple graphs was considered before by Aharoni and Berger. We also disprove one of their conjectures regarding the lower bound on the number of colors one needs in the conjecture of Barát, Gyárfás, and Sárközy for the class of simple graphs. Our methods are inspired by the randomized algorithm proposed by Gao, Ramadurai, Wanless, and Wormald to find a rainbow matching of size q in a graph that is properly edge-colored with q colors, where each color class contains q + o(q) edges. We consider a modified version of their algorithm, with which we are able to prove a generalization of their statement with a slightly better error term in o(q). As a by-product of our techniques, we obtain a new asymptotic version of the Brualdi-Ryser-Stein Conjecture.

show abstract

“…When the matchings have size 2n then it is easy to see that the conclusion of the conjecture is true (by greedily choosing disjoint edges one at a time). 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 [8] improved this to 5n/3. Clemens and Ehrenmüller [5] further improved this to 3n/2 + o(n) which is currently the best known bound.…”

Section: Conjecture 13 (Aharoni and Berger [1]mentioning

confidence: 99%

Rainbow matchings and connectedness of coloured graphs

Pokrovskiy

2015

Electronic Notes in Discrete Mathematics

View full text Add to dashboard Cite

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. When the matchings are all edge-disjoint and perfect, an approximate version of this conjecture follows from a theorem of Häggkvist and Johansson which implies the conjecture when the matchings have size at least n + o(n). Here we'll discuss a proof of this conjecture in the case when the matchings have size n + o(n) and are all edge-disjoint (but not necessarily perfect). The proof involves studying connectedness in coloured, directed graphs. The notion of connectedness that we introduce is new, and perhaps of independent interest.

show abstract

Large matchings in bipartite graphs have a rainbow matching

Cited by 20 publications

References 12 publications

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

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

Rainbow matchings in edge-colored simple graphs

Rainbow matchings and connectedness of coloured graphs

Contact Info

Product

Resources

About