Related papers: Counterexamples on matchings in hypergraphs and fu…
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 $\M$…
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…
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 $H_{1}$, \cdots, $H_{s}$ are properly-colored $k$-uniform hypergraphs on $n$ vertices,…
Given a multi-hypergraph $G$ that is edge-colored into color classes $E_1, \ldots, E_n$, a full rainbow matching is a matching of $G$ that contains exactly one edge from each color class $E_i$. One way to guarantee the existence of a full…
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…
We show that if a multigraph $G$ with maximum edge-multiplicity of at most $\frac{\sqrt{n}}{\log^2 n}$, is edge-coloured by $n$ colours such that each colour class is a disjoint union of cliques with at least $2n + o(n)$ vertices, then it…
A rainbow matching in an edge-coloured graph is a matching such that its edges have distinct colours. We show that every properly edge-coloured graph $G$ with $|G| \ge (9\delta(G) -5)/2$ has a rainbow matching of size $\delta(G)$, improving…
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…
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…
Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges and colors in $G$, respectively. A subgraph $H$ is called rainbow if $c(H)=e(H)$. Li et al. (European J. Combin., 36 (2014), 453-459) proved that every…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many…
Let $g(n)$ be the least number such that every collection of $n$ matchings, each of size at least $g(n)$, in a bipartite graph, has a full rainbow matching. Aharoni and Berger \cite{AhBer} conjectured that $g(n)=n+1$ for every $n>1$. This…
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…
An edge-coloring of a complete graph with a set of colors $C$ is called completely balanced if any vertex is incident to the same number of edges of each color from $C$. Erd\H{o}s and Tuza asked in $1993$ whether for any graph $F$ on $\ell$…
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi that applies to almost optimally bounded colourings. A…
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…
An edge-colouring of a graph $G$ can fail to be rainbow for two reasons: either it contains a monochromatic cherry (a pair of incident edges), or a monochromatic matching of size two. A colouring is a proper colouring if it forbids the…
A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive…
A {\it 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…