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 χ ˆ ′ ( 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 χ ˆ ′ ( G ) < t ( t + 1 ) n ln n , and examples exist with χ ˆ ′ ( G ) ≥ t 2 ( n − 1 ) . The rainbow domination number, written γ ˆ ( 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) γ ˆ ( G ) ≤ 1 + ln k k n (where k = δ ( G ) t + 1 ), and (2) γ ˆ ( G ) ≤ t t + 1 n when G has no isolated vertices. We also characterize the edge-colored graphs achieving equality in the latter bound.