Rainbow Matching Research Articles

Rainbow Matchings and Rainbow Connectedness

Aharoni and Berger conjectured that every collection of $n$ matchings of size $n+1$ in a bipartite graph contains a rainbow matching of size $n$. This conjecture is related to 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. The conjecture is known to hold when the matchings are much larger than $n+1$. The best bound is currently due to Aharoni, Kotlar, and Ziv who proved the conjecture when the matchings are of size at least $3n/2+1$. When the matchings are all edge-disjoint and perfect, 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 $\phi n+o(n)$ where $\phi\approx 1.618$ 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.

A Rainbow r-Partite Version of the Erdős–Ko–Rado Theorem

Let [n]r be the complete r-partite hypergraph with vertex classes of size n. It is an easy exercise to show that every set of more than (k−1)nr−1 edges in [n]r contains a matching of size k. We conjecture the following rainbow version of this observation: if F1,F2,. . .,Fk ⊆ [n]r are of size larger than (k−1)nr−1 then there exists a rainbow matching, that is, a choice of disjoint edges fi ∈ Fi. We prove this conjecture for r=2 and r=3.

Rainbow Matchings and Algebras of Sets

Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number $\mathfrak v = \mathfrak v(n)$ such that, if $A_1, \ldots, A_n$ are $n$ equivalence relations on a common finite ground set $X$, such that for each $i$ there are at least $\mathfrak v$ elements of $X$ that belong to $A_i$-equivalence classes of size larger than $1$, then $X$ has a rainbow matching---a set of $2n$ distinct elements $a_1, b_1, \ldots, a_n, b_n$, such that $a_i$ is $A_i$-equivalent to $b_i$ for each $i$? Grinblat has shown that $\mathfrak v(n) \le 10n/3 + O(\sqrt{n})$. He asks whether $\mathfrak v(n) = 3n-2$ for all $n\ge 4$. In this paper we improve the upper bound (for all large enough $n$) to $\mathfrak v(n) \le 16n/5 + O(1)$.

On a conjecture of Stein

Stein (Pac J Math 59:567–575, 1975) proposed the following conjecture: if the edge set of $$K_{n,n}$$ is partitioned into n sets, each of size n, then there is a partial rainbow matching of size $$n-1$$ . He proved that there is a partial rainbow matching of size $$n(1-\frac{D_n}{n!})$$ , where $$D_n$$ is the number of derangements of [n]. This means that there is a partial rainbow matching of size about $$(1- \frac{1}{e})n$$ . Using a topological version of Hall’s theorem we improve this bound to $$\frac{2}{3}n$$ .

Representation of Large Matchings in Bipartite Graphs

Let $f(n)$ be the smallest number such that every collection of $n$ matchings, each of size at least $f(n)$, in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi, and Stein, Aharoni and Berger [Electron. J. Combin., 16 (2009), R119] conjectured that $f(n)=n+1$ for every $n>1$. Clemens and Ehrenmüller proved that $f(n) \le \frac{3}{2}n +o(n)$. We show that the $o(n)$ term can be reduced to a constant, namely, $f(n) \le \lceil \frac{3}{2}n \rceil+1$.

Rainbow matchings in bipartite multigraphs

Suppose that k is a non-negative integer and a bipartite multigraph G is the union of $$\begin{aligned} N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1) \end{aligned}$$ matchings \(M_1,\dots ,M_N\), each of size n. We show that G has a rainbow matching of size \(n-k\), i.e. a matching of size \(n-k\) with all edges coming from different \(M_i\)’s. Several choices of the parameter k relate to known results and conjectures.

On rainbow matchings in bipartite graphs

We present recent results regarding rainbow matchings in bipartite graphs. Using topological methods we address a known conjecture of Stein and show that if Kn,n is partitioned into n sets of size n, then a partial rainbow matching of size 2n/3 exists. We generalize a result of Cameron and Wanless and show that for any n matchings of size n in a bipartite graph with 2n vertices there exists a full matching intersecting each matching at most twice. We show that any n matchings of size approximately 3n/2 have a rainbow matching of size n. Finally, we show the uniqueness of the extreme case for a theorem of Drisko and provide a generalization of Drisko's theorem.

On sets not belonging to algebras and rainbow matchings in graphs

Motivated by a question of Grinblat, we study the minimal number v(n) that satisfies the following. If A1,…,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 Ai are nontrivial, then A1,…,An contain a rainbow matching, i.e. there exist 2n distinct elements x1,y1,…,xn,yn∈X with xi∼Aiyi 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 Nivasch 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).

Existence of rainbow matchings in strongly edge-colored graphs

The famous Ryser Conjecture states that there is a transversal of size n in a latin square of odd order n, which is equivalent to finding a rainbow matching of size n in a properly edge-colored Kn,n when n is odd. Let δ denote the minimum degree of a graph. In 2011, Wang proposed a more general question to find a function f(δ) (f(δ)≥2δ+1) such that for each properly edge-colored graph of order f(δ), there exists a rainbow matching of size δ. The best bound so far is f(δ)≤3.5δ+2 due to Lo. Babu et al. considered this problem in strongly edge-colored graphs in which each path of length 3 is rainbow. They proved that if G is a strongly edge-colored graph of order at least 2⌊3δ4⌋, then G has a rainbow matching of size ⌊3δ4⌋. They proposed an interesting question: Is there a constant c greater than 34 such that every strongly edge-colored graph G has a rainbow matching of size at least cδ if |V(G)|≥2⌊cδ⌋? Clearly, c≤1. We prove that if G is a strongly edge-colored graph with minimum degree δ and order at least 2δ+2, then G has a rainbow matching of size δ.

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

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 $\left(\frac 3 2 + o(1)\right) n$ 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 by Aharoni, Charbit and Howard, and Kotlar and Ziv.

Rainbow matchings and algebras of sets

Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number v=v(n) such that, if A1,…,An are n equivalence relations on a common finite ground set X, such that for each i there are at least v elements of X that belong to Ai-equivalence classes of size larger than 1, then X has a rainbow matching—a set of 2n distinct elements a1,b1,…,an,bn, such that ai is Ai-equivalent to bi for each i?Grinblat has shown that v(n)≤10n/3+O(nn). He asks whether v(n)=3n−2 for all n≥4. In this paper we improve the upper bound (for all large enough n) to v(n)≤16n/5+O(1).

Rainbow matchings and connectedness of coloured graphs

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.

Rainbow matchings and Hamilton cycles in random graphs

AbstractLet be drawn uniformly from all m‐edge, k‐uniform, k‐partite hypergraphs where each part of the partition is a disjoint copy of . We let be an edge colored version, where we color each edge randomly from one of colors. We show that if and where K is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if n is even and where K is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in . Here denotes a random edge coloring of with n colors. When n is odd, our proof requires for there to be a rainbow Hamilton cycle. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 503–523, 2016

Existences of rainbow matchings and rainbow matching covers

Let G be an edge-coloured graph. A rainbow subgraph in G is a subgraph such that its edges have distinct colours. The minimum colour degree δc(G) of G is the smallest number of distinct colours on the edges incident with a vertex of G. We show that every edge-coloured graph G on n≥7k/2+2 vertices with δc(G)≥k contains a rainbow matching of size at least k, which improves the previous result for k≥10.Let Δmon(G) be the maximum number of edges of the same colour incident with a vertex of G. We also prove that if t≥11 and Δmon(G)≤t, then G can be edge-decomposed into at most ⌊tn/2⌋ rainbow matchings. This result is sharp and improves a result of LeSaulnier and West.

Rainbow matchings in strongly edge-colored graphs

A rainbow matching of an edge-colored graph G is a matching in which no two edges have the same color. There have been several studies regarding the maximum size of a rainbow matching in a properly edge-colored graph G in terms of its minimum degree δ(G). Wang (2011) asked whether there exists a function f such that a properly edge-colored graph G with at least f(δ(G)) vertices is guaranteed to contain a rainbow matching of size δ(G). This was answered in the affirmative later: the best currently known function Lo and Tan (2014) is f(k)=4k−4, for k≥4 and f(k)=4k−3, for k≤3. Afterwards, the research was focused on finding lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs with fewer than 4δ(G)−4 vertices. Strong edge-coloring of a graph G is a restriction of proper edge-coloring where every color class is required to be an induced matching, instead of just being a matching. In this paper, we give lower bounds for the size of a maximum rainbow matching in a strongly edge-colored graph G in terms of δ(G). We show that for a strongly edge-colored graph G, if |V(G)|≥2⌊3δ(G)4⌋, then G has a rainbow matching of size ⌊3δ(G)4⌋, and if |V(G)|<2⌊3δ(G)4⌋, then G has a rainbow matching of size ⌊|V(G)|2⌋. In addition, we prove that if G is a strongly edge-colored graph that is triangle-free, then it contains a rainbow matching of size at least δ(G).

Rainbow numbers for matchings in plane triangulations

Given two graphs G and H, let f(G,H) denote the maximum number c for which there is a way to color the edges of G with c colors such that every subgraph H of G has at least two edges of the same color. Equivalently, any edge-coloring of G with at least rb(G,H)=f(G,H)+1 colors contains a rainbow copy of H, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number rb(G,H) is called the rainbow number of Hwith respect to G. If G is a complete graph Kn, the numbers f(Kn,H) and rb(Kn,H) are called anti-Ramsey numbers and rainbow numbers, respectively.In this paper we will study the existence of rainbow matchings in plane triangulations. Denote by kK2 a matching of size k and Tn the class of all plane triangulations of order n. The rainbow numberrb(Tn,kK2) is the minimum number of colors c such that, if kK2⊆Tn∈Tn, then any edge-coloring of Tn with at least c colors contains a rainbow copy of kK2. We give lower and upper bounds on rb(Tn,kK2) for all k≥3 and n≥2k. Furthermore, we obtain the exact values of rb(Tn,kK2) for 2≤k≤4 and n≥2k.

On a Generalization of the Ryser-Brualdi-Stein Conjecture

A rainbow matching for not necessarily distinct sets F1,',Fk of hypergraph edges is a matching consisting of k edges, one from each Fi. The aim of the article is twofold-to put order in the multitude of conjectures that relate to this concept some first presented here, and to prove partial results on one of the central conjectures.

Rainbow matchings and cycle-free partial transversals of Latin squares

In this paper we show that properly edge-colored graphs G with |V(G)|≥4δ(G)−3 have rainbow matchings of size δ(G); this gives the best known bound for a recent question of Wang. We also show that properly edge-colored graphs G with |V(G)|≥2δ(G) have rainbow matchings of size at least δ(G)−2δ(G)2/3. This result extends (with a weaker error term) the well-known result that a factorization of the complete bipartite graph Kn,n has a rainbow matching of size n−o(n), or equivalently that every Latin square of order n has a partial transversal of size n−o(n) (an asymptotic version of the Ryser–Brualdi conjecture). In this direction we also show that every Latin square of order n has a cycle-free partial transversal of size n−o(n).

How Many Colors Guarantee a Rainbow Matching?

Given a coloring of the edges of a multi-hypergraph, a rainbow $t$-matching is a collection of $t$ disjoint edges, each having a different color. In this note we study the problem of finding a rainbow $t$-matching in an $r$-partite $r$-uniform multi-hypergraph whose edges are colored with $f$ colors such that every color class is a matching of size $t$. This problem was posed by Aharoni and Berger, who asked to determine the minimum number of colors which guarantees a rainbow matching. We improve on the known upper bounds for this problem for all values of the parameters. In particular for every fixed $r$, we give an upper bound which is polynomial in $t$, improving the superexponential estimate of Alon. Our proof also works in the setting not requiring the hypergraph to be $r$-partite.

Complexity results for rainbow matchings

A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem.We show, among others, that•max rainbow matching can be approximated by a polynomial algorithm with approximation ratio 2/3−ε.•max rainbow matching is APX-complete, even when restricted to properly edge-colored linear forests without a 5-vertex path, and is solvable in polynomial time for edge-colored forests without a 4-vertex path.•max rainbow matching is APX-complete, even when restricted to properly edge-colored trees without an 8-vertex path, and is solvable in polynomial time for edge-colored trees without a 7-vertex path.•max rainbow matching is APX-complete, even when restricted to properly edge-colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths.We also address the parameterized complexity of the problem.