相关论文: Large matchings in bipartite graphs have a rainbow…

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…

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…

Aharoni and Berger conjectured that in every proper edge-colouring of a bipartite multigraph by $n$ colours with at least $n+1$ edges of each colour there is a rainbow matching using every colour. This conjecture generalizes a longstanding…

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…

A folklore result on matchings in graphs states that if $G$ is a bipartite graph whose vertex classes $A$ and $B$ each have size $n$, with $\mathrm{deg}(u) \geq a$ for every $u \in A$ and $\mathrm{deg}(v) \geq b$ for every $v \in B$, then…

Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1)$$ matchings $M_1,\dots,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size…

Let $G$ be a simple graph that is properly edge coloured with $m$ colours and let $\M=\{M_1,\ldots, M_m\}$ be the set of $m$ matchings induced by the colours in $G$. Suppose that $m\le n-n^{c}$, where $c>9/10$, and every matching in $\M$…

By a theorem of Drisko, any $2n-1$ matchings of size $n$ in a bipartite graph have a partial rainbow matching of size $n$. Inspired by discussion of Bar\'at, Gy\'arf\'as and S\'ark\"ozy, we conjecture that if $n$ is odd then the same is…

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…

Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to…

A graph $G$ whose edges are coloured (not necessarily properly) contains a full rainbow matching if there is a matching $M$ that contains exactly one edge of each colour. We refute several conjectures on matchings in hypergraphs and full…

We prove that any family $E_1, \ldots , E_{\lceil rn \rceil}$ of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$, has a rainbow fractional matching of size $n$ (that is, a…

A subgraph of an edge-colored graph is called \emph{rainbow} if all of its edges have distinct colors. There has been much research on the topic of finding a large rainbow matching in a properly edge-colored graph, where a proper…

A natural question, inspired by the famous Ryser-Brualdi-Stein Conjecture, is to determine the largest positive integer $g(r,n)$ such that every collection of $n$ matchings, each of size $n$, in an $r$-partite $r$-uniform hypergraph…

We prove that, for positive integers $n,a_1, a_2, a_3$ satisfying $a_1+a_2+a_3 = n-1$, it holds that any bipartite graph $G$ which is the union of three perfect matchings $M_1$, $M_2$, and $M_3$ on $2n$ vertices contains a matching $M$ such…

Many well-known problems in Combinatorics can be reduced to finding a large rainbow structure in a certain edge-coloured multigraph. Two celebrated examples of this are Ringel's tree packing conjecture and Ryser's conjecture on transversals…

A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of…

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…

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by $n$ colours with at least $n + 1$ edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider…

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…