Rainbow perfect matchings in r-partite graph structures

A matching M in an edge–colored (hyper)graph is rainbow if each pair of edges in M have distinct colors. We extend the result of Erdos and Spencer on the existence of rainbow perfect matchings in the complete bipartite graph Kn,n to complete bipartite multigraphs, dense regular bipartite graphs and complete r-partite r-uniform hypergraphs. The proof of the results use the Lopsided version of the Local Lovász Lemma.Peer ReviewedPostprint (author's final draft

Flattening rank and its combinatorial applications

Given a $d$-dimensional tensor $T:A_1\times\dots\times A_d\rightarrow \mathbb{F}$ (where $\mathbb{F}$ is a field), the $i$-flattening rank of $T$ is the rank of the matrix whose rows are indexed by $A_{i}$, columns are indexed by $B_{i}=A_1\times\dots\times A_{i-1}\times A_{i+1}\times\dots\times A_{d}$ and whose entries are given by the corresponding values of $T$. The max-flattening rank of $T$ is defined as $\text{mfrank}(T)=\max_{i\in [d]}\text{frank}_{i}(T)$. A tensor $T:A^{d}\rightarrow\mathbb{F}$ is called semi-diagonal, if $T(a,\dots,a)\neq 0$ for every $a\in A$, and $T(a_{1},\dots,a_{d})=0$ for every $a_{1},\dots,a_{d}\in A$ that are all distinct. In this paper we prove that if $T:A^{d}\rightarrow\mathbb{F}$ is semi-diagonal, then $\text{mfrank}(T)\geq \frac{|A|}{d-1}$, 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 $\mathcal{H}$ are colored with $z$ colors such that each colorclass is a matching of size $t$, then $\mathcal{H}$ contains a rainbow matching of size $t$ provided $z>(t-1)\binom{rt}{r}$. This improves previous results of Alon and Glebov, Sudakov and Szab\'o

Rainbow matchings in hypergraphs

A rainbow matching in an edge colored multihypergraph is a matching consisting of edges with pairwise distinct colors. In this master thesis we give an overview of the results about having a rainbow matching in edge-colored bipartite graphs and edge colored r-partite r-uniform hypergraphs. Having in mind the techniques that are used in the last results we aplied them and we get some new approaches using the Local Lovasz Lemma

How Many Colors Guarantee a Rainbow Matching?

ISSN:1097-1440ISSN:1077-892