Full rainbow matchings in graphs and hypergraphs

Let G be a simple graph that is properly edge-coloured with m colours and let \[\mathcal{M} = \{ {M_1},...,{M_m}\} \] be the set of m matchings induced by the colours in G. Suppose that \[m \leqslant n - {n^c}\] , where \[c > 9/10\] , and every matching in \[\mathcal{M}\] has size n. Then G contains a full rainbow matching, i.e. a matching that contains exactly one edge from M i for each \[1 \leqslant i \leqslant m\] . This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalization of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs. Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

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 $s=r$.

Rainbow Matchings of size $m$ in Graphs with Total Color Degree at least $2mn$

The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper generalizes these problems to edge colored graphs with given total color degree. In particular, we find that if a graph $G$ has total color degree $2mn$ and satisfies some other properties, then $G$ contains a matching of size $m$. These other properties include $G$ being triangle-free, $C_4$-free, properly colored, or large enough.

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.

Large rainbow matchings in semi-strong edge-colorings of graphs

The lower bounds for the size of maximum rainbow matching in properly edge-colored graphs have been studied deeply during the last decades. An edge-coloring of a graph [Formula: see text] is called a strong edge-coloring if each path of length at most three is rainbow. Clearly, the strong edge-coloring is a natural generalization of the proper one. Recently, Babu et al. considered the problem in the strongly edge-colored graphs. In this paper, we introduce a semi-strong edge-coloring of graphs and consider the existence of large rainbow matchings in it.

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.