4 research outputs found
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
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
Colored Saturation Parameters for Bipartite Graphs
Let F and H be fixed graphs and let G be a spanning subgraph of H. G is an F-free subgraph of H if F is not a subgraph of G. We say that G is an F-saturated subgraph of H if G is F-free and for any edge e in E(H)-E(G), F is a subgraph of G+e. The saturation number of F in K_{n,n}, denoted sat(K_{n,n}, F), is the minimum size of an F-saturated subgraph of K_{n,n}. A t-edge-coloring of a graph G is a labeling f: E(G) to [t], where [t] denotes the set { 1, 2, ..., t }. The labels assigned to the edges are called colors. A rainbow coloring is a coloring in which all edges have distinct colors. Given a family F of edge-colored graphs, a t-edge-colored graph H is (F, t)-saturated if H contains no member of F but the addition of any edge in any color completes a member of F. In this thesis we study the minimum size of ( F,t)-saturated subgraphs of edge-colored complete bipartite graphs. Specifically we provide bounds on the minimum size of these subgraphs for a variety of families of edge-colored bipartite graphs, including monochromatic matchings, rainbow matchings, and rainbow stars