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

Large rainbow matchings in large graphs

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 (namely, $n\geq 4.25k^2$), then each $n$-vertex graph $G$ with minimum color degree at least $k$ contains a rainbow matching of size at least $k$

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 $\delta^c(G)$ 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 $n\geq 7k/2+2$ vertices with $\delta^c(G) \geq k$ contains a rainbow matching of size at least $k$, which improves the previous result for $k \ge 10$. Let $\Delta_{\text{mon}}(G)$ be the maximum number of edges of the same colour incident with a vertex of $G$. We also prove that if $t \ge 11$ and $\Delta_{\text{mon}}(G) \le t$, then $G$ can be edge-decomposed into at most $\lfloor tn/2 \rfloor$ rainbow matchings. This result is sharp and improves a result of LeSaulnier and West