Rainbow Matching Of Size Research Articles

Large Rainbow Matchings in Edge-Colored Graphs with Given Average Color Degree

A rainbow matching in an edge-colored graph is a matching in which no two edges have the same color. The color degree of a vertex v is the number of distinct colors on edges incident to v. Kritschgau [Electron. J. Combin. 27(3), 2020] studied the existence of rainbow matchings in edge-colored graph G with average color degree at least 2k, and proved some sufficient conditions for a rainbow marching of size k in G. The sufficient conditions include that $$|V(G)|\ge 12k^2+4k$$ , or G is a properly edge-colored graph with $$|V(G)|\ge 8k$$ . In this paper, we show that every edge-colored graph G with $$|V(G)|\ge 4k-4$$ and average color degree at least $$2k-1$$ contains a rainbow matching of size k. In addition, we also prove that every strongly edge-colored graph G with average degree at least $$2k-1$$ contains a rainbow matching of size at least k. The bound is sharp for complete graphs.

Leray numbers of complexes of graphs with bounded matching number

Given a graph G on the vertex set V, the non-matching complex of G, denoted by NMk(G), is the family of subgraphs G′⊂G whose matching number ν(G′) is strictly less than k. As an attempt to extend the result by Linusson, Shareshian and Welker on the homotopy types of NMk(Kn) and NMk(Kr,s) to arbitrary graphs G, we show that (i) NMk(G) is (3k−3)-Leray, and (ii) if G is bipartite, then NMk(G) is (2k−2)-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex NMk(G), which vanishes in all dimensions d≥3k−4, and all dimensions d≥2k−3 when G is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes a result by Aharoni, Berger, Chudnovsky, Howard and Seymour: Let E1,…,E3k−2 be non-empty edge subsets of a graph and suppose that ν(Ei∪Ej)≥k for every i≠j. Then E=⋃Ei has a rainbow matching of size k. Furthermore, the number of edge sets Ei can be reduced to 2k−1 when E is the edge set of a bipartite graph.

Cooperative Conditions for the Existence of Rainbow Matchings

Let $k>1$, and let $\mathcal{F}$ be a family of $2n+k-3$ non-empty sets of edges in a bipartite graph. If the union of every $k$ members of $\mathcal{F}$ contains a matching of size $n$, then there exists an $\mathcal{F}$-rainbow matching of size $n$. Replacing $2n+k-3$ by $2n+k-2$, the result is true also for $k=1$, and it can be proved (for all $k$) both topologically and by a relatively simple combinatorial argument. The main effort is in gaining the last $1$, which makes the result sharp.

Rainbow Paths and Large Rainbow Matchings

A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\frac{2}{3}n-1$, improving on the trivial $\frac{1}{2}n$, and an analogous result for hypergraphs. For $\{C_3,C_5\}$-free graphs and for disjoint matchings we obtain a lower bound of $\frac{3n}{4}-O(1)$. We also discuss a conjecture on rainbow alternating paths, that if true would yield a lower bound of $n-\sqrt{2n}$. We prove the non-alternating (ordinary paths) version of this conjecture.

Flattening rank and its combinatorial applications

Given a d-dimensional tensor T:A1×…×Ad→F (where F is a field), the i-flattening rank of T is the rank of the matrix whose rows are indexed by Ai, columns are indexed by Bi=A1×…×Ai−1×Ai+1×…×Ad and whose entries are given by the corresponding values of T. The max-flattening rank of T is defined as mfrank(T)=maxi∈[d]⁡franki(T). A tensor T:Ad→F is called semi-diagonal, if T(a,…,a)≠0 for every a∈A, and T(a1,…,ad)=0 for every a1,…,ad∈A that are all distinct. In this paper we prove that if T:Ad→F is semi-diagonal, then mfrank(T)≥|A|d−1, and this bound is the best possible.We give several applications of this result, including a generalization of the celebrated Frankl-Wilson theorem on forbidden intersections. Also, addressing a conjecture of Aharoni and Berger, we show that if the edges of an r-uniform multi-hypergraph H are colored with z colors such that each color class is a matching of size t, then H contains a rainbow matching of size t provided z>(t−1)(rtr). This improves previous results of Alon and Glebov, Sudakov, and Szabó.

Badges and rainbow matchings

Drisko proved that 2n−1 matchings of size n in a bipartite graph have a rainbow matching of size n. For general graphs it is conjectured that 2n matchings suffice for this purpose (and that 2n−1 matchings suffice when n is even). The known graphs showing sharpness of this conjecture for n even are called badges. We improve the previously best known bound from 3n−2 to 3n−3, using a new line of proof that involves analysis of the appearance of badges. We also prove a “cooperative” generalization: for t>0 and n≥3, any 3n−4+t sets of edges, the union of every t of which contains a matching of size n, have a rainbow matching of size n.

The Maximum Number of Cliques in Hypergraphs without Large Matchings

Let $[n]$ denote the set $\{1, 2, \ldots, n\}$ and $\mathcal{F}^{(r)}_{n,k,a}$ be an $r$-uniform hypergraph on the vertex set $[n]$ with edge set consisting of all the $r$-element subsets of $[n]$ that contains at least $a$ vertices in $[ak+a-1]$. For $n\geq 2rk$, Frankl proved that $\mathcal{F}^{(r)}_{n,k,1}$ maximizes the number of edges in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$. Huang, Loh and Sudakov considered a multicolored version of the Erd\H{o}s matching conjecture, and provided a sufficient condition on the number of edges for a multicolored hypergraph to contain a rainbow matching of size $k$. In this paper, we show that $\mathcal{F}^{(r)}_{n,k,a}$ maximizes the number of $s$-cliques in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$ for sufficiently large $n$, where $a=\lfloor \frac{s-r}{k} \rfloor+1$. We also obtain a condition on the number of $s$-clques for a multicolored $r$-uniform hypergraph to contain a rainbow matching of size $k$, which reduces to the condition of Huang, Loh and Sudakov when $s=r$.

Large rainbow matchings in general graphs

By a theorem of Drisko, any 2n−1 matchings of size n in a bipartite graph have a rainbow matching of size n. Inspired by results and discussion of Barát, Gyárfás and Sárközy, we conjecture that if n is odd then the same is true also in general graphs, and that if n is even then 2n matchings of size n suffice. We prove that any 3n−2 matchings of size n have a rainbow matching of size n.

Two problems on matchings in set families – In the footsteps of Erdős and Kleitman

The families F1,…,Fs⊂2[n] are called q-dependent if there are no pairwise disjoint F1∈F1,…,Fs∈Fs satisfying |F1∪…∪Fs|≤q. We determine max⁡|F1|+…+|Fs| for all values n≥q,s≥2. The result provides a far-reaching generalization of an important classical result of Kleitman.The well-known Erdős Matching Conjecture suggests the largest size of a family F⊂([n]k) with no s pairwise disjoint sets. After more than 50 years its full solution is still not in sight. In the present paper we provide a Hilton–Milner-type stability theorem for the Erdős Matching Conjecture in a relatively wide range, in particular, for n≥(2+o(1))sk with o(1) depending on s only. This is a considerable improvement of a classical result due to Bollobás, Daykin and Erdős.We apply our results to advance in the following anti-Ramsey-type problem, proposed by Özkahya and Young. Let ar(n,k,s) be the minimum number x of colors such that in any coloring of the k-element subsets of [n] with x (non-empty) colors there is a rainbow matching of size s, that is, s sets of different colors that are pairwise disjoint. We prove a stability result for the problem, which allows to determine ar(n,k,s) for all k≥3 and n≥sk+(s−1)(k−1). Some other consequences of our results are presented as well.

Rainbow Matchings in Properly-Colored Hypergraphs

A hypergraph $H$ is properly colored if for every vertex $v\in V(H)$, all the edges incident to $v$ have distinct colors. In this paper, we show that if $H_{1}, \ldots, H_{s}$ are properly-colored $k$-uniform hypergraphs on $n$ vertices, where $n\geq3k^{2}s$, and $e(H_{i})>{{n}\choose {k}}-{{n-s+1}\choose {k}}$, then there exists a rainbow matching of size $s$, containing one edge from each $H_i$. This generalizes some previous results on the Erdős Matching Conjecture.

A note on rainbow matchings in strongly edge-colored graphs

The Ryser Conjecture which states that there is a transversal of size n in a Latin square of odd order n is equivalent to finding a rainbow matching of size n in a properly edge-colored Kn,n using n colors when n is odd. Let δ be the minimum degree of a graph. Wang proposed a more general question to find a function f(δ) such that every properly edge-colored graph of order f(δ) contains a rainbow matching of size δ, which currently has the best bound of f(δ)≤3.5δ+2 by Lo. Babu, Chandran and Vaidyanathan investigated Wang’s question under a stronger color condition. A strongly edge-colored graph is a properly edge-colored graph in which every monochromatic subgraph is an induced matching. Wang, Yan and Yu proved that every strongly edge-colored graph of order at least 2δ+2 has a rainbow matching of size δ. In this note, we extend this result to graphs of order at least 2δ+1.

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.

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.

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.

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 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).

A NOTE ON RAINBOW MATCHINGS

Let G be a d-regular properly edge-colored graph such that the edges with the same color induce a 1-factor of G. A rainbow matching of G is a matching in which no two edges have the same color. We show that if the order n of G is at least 3.79d, then G has a rainbow matching of size d.

A Note on Large Rainbow Matchings in Edge-coloured Graphs

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on n vertices with minimum colour degree at least k contains a rainbow matching of size at least k, provided $${n\geq \frac{17}{4}k^2}$$ . In this paper, we show that n ≥ 4k − 4 is sufficient for k ≥ 4.

Rainbow Matchings of Size $\delta(G)$ in Properly Edge-Colored Graphs

A rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function $f(\delta)$ such that a properly edge-colored graph $G$ with minimum degree $\delta$ and order at least $f(\delta)$ must have a rainbow matching of size $\delta$. We answer this question in the affirmative; an extremal approach yields that $f(\delta) = 98\delta/23&lt; 4.27\delta$ suffices. Furthermore, we give an $O(\delta(G)|V(G)|^2)$-time algorithm that generates such a matching in a properly edge-colored graph of order at least $6.5\delta$.