Rainbow edge-coloring and rainbow domination

Let G be an edge-colored graph with n vertices. A rainbow subgraph is a subgraph whose edges have distinct colors. The rainbow edge-chromatic number of G, written χˆ′(G), is the minimum number of rainbow matchings needed to cover E(G). An edge-colored graph is t-tolerant if it contains no monochromatic star with t+1 edges. If G is t-tolerant, then χˆ′(G)