Rainbow numbers for small stars with one edge added

A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f(n, H) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H . The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of Kn with rb(n, H) number of colors contains a rainbow copy of H . Certainly rb(n, H) = f(n, H) + 1. Anti-Ramsey numbers were introduced by Erdös et al. [5] and studied in numerous papers. We show that rb(n, K1,4 + e) = n + 2 in all nontrivial cases.