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