The minimum size of -rainbow connected graphs of given order

Let t ( n , k ) denote the minimum number of edges of a k -rainbow connected graph G of order n , that is a graph G for which a coloring of the edges with at most k colors exists such that any two vertices of G are connected by a path with edges of pairwise different colors. A general upper bound of t ( n , k ) and the exact value of t ( n , 3 ) are given.