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 χ̂′(G), is the minimum number of rainbow matchings needed to cover E(G). An edgecolored graph is t-tolerant if it contains no monochromatic star with t+1 edges. If G is t-tolerant, then χ̂′(G) < t(t+ 1)n lnn, and examples exist with χ̂′(G) ≥ t 2(n− 1). The rainbow domination number, written γ̂(G), is the minimum number of disjoint rainbow stars needed to cover V (G). For t-tolerant edge-colored n-vertex graphs, we generalize classical bounds on the domination number: (1) γ̂(G) ≤ 1+ln k k n (where k = δ(G) t + 1), and (2) γ̂(G) ≤ t t+1n when G has no isolated vertices. We also characterize the edgecolored graphs achieving equality in the latter bound.