Rainbow edge-coloring and rainbow domination

Abstract

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)<t(t+1)nlnn, and examples exist with χˆ′(G)≥t2(n−1). The rainbow domination number, written γˆ(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) γˆ(G)≤1+lnkkn (where k=δ(G)t+1), and (2) γˆ(G)≤tt+1n when G has no isolated vertices. We also characterize the edge-colored graphs achieving equality in the latter bound.