Rainbow matchings in bipartite multigraphs

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 $n-k$, i.e. a matching of size $n-k$ with all edges coming from different $M_i$'s. Several choices of parameters relate to known results and conjectures

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

Rainbow matchings in properly-coloured multigraphs

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 the same question without the bipartiteness assumption. We show that in any multigraph with edge multiplicities $o(n)$ that is properly edge-coloured by $n$ colours with at least $n + o(n)$ edges of each colour there must be a matching of size $n-O(1)$ that uses each colour at most once.Comment: 7 page

Rainbow Hamilton cycles in random regular graphs

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n tends to infinity, provided d is at least 8.Comment: 16 page