592 research outputs found
Existences of rainbow matchings and rainbow matching covers
Let be an edge-coloured graph. A rainbow subgraph in is a subgraph
such that its edges have distinct colours. The minimum colour degree
of is the smallest number of distinct colours on the edges
incident with a vertex of . We show that every edge-coloured graph on
vertices with contains a rainbow matching
of size at least , which improves the previous result for .
Let be the maximum number of edges of the same
colour incident with a vertex of . We also prove that if and
, then can be edge-decomposed into at most
rainbow matchings. This result is sharp and improves a
result of LeSaulnier and West
Rainbow Matchings and Hamilton Cycles in Random Graphs
Let be drawn uniformly from all -uniform, -partite
hypergraphs where each part of the partition is a disjoint copy of . We
let HP^{(\k)}_{n,m,k} be an edge colored version, where we color each edge
randomly from one of \k colors. We show that if \k=n and where
is sufficiently large then w.h.p. there is a rainbow colored perfect
matching. I.e. a perfect matching in which every edge has a different color. We
also show that if is even and where is sufficiently large
then w.h.p. there is a rainbow colored Hamilton cycle in . Here
denotes a random edge coloring of with colors.
When is odd, our proof requires m=\om(n\log n) for there to be a rainbow
Hamilton cycle.Comment: We replaced graphs by k-uniform hypergraph
Rainbow matchings in bipartite multigraphs
Suppose that is a non-negative integer and a bipartite multigraph is
the union of matchings
, each of size . We show that has a rainbow matching of
size , i.e. a matching of size with all edges coming from different
's. Several choices of parameters relate to known results and conjectures
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