An edge-colored tree T is a rainbow tree if no two edges of T are assigned the same color. Let G be a nontrivial connected graph of order n and let k be an integer with 2 <= k <= n. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S subset of V(T). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G. For every two integers k and n >= 3 with 3 <= k <= n, the k-rainbow index of a unicyclic graph of order n is determined. For a set S of vertices in a connected graph G of order n, a collection {T-1, T-2, ..., T-l} of trees in G is said to be internally disjoint connecting S if these trees are pairwise edge-disjoint and V(T-i) boolean AND V(T-j) = S for every pair i, j of distinct integers with 1 <= i, j <= l. For an integer k with 2 <= k <= n, the k-connectivity kappa(k)(G) of G is the greatest positive integer l for which G contains at least l internally disjoint trees connecting S for every set S of k vertices of G. It is shown that kappa(k) (K-n) = n - [k/2] for every pair k, n of integers with 2 <= k <= n. For a nontrivial connected graph G of order n and for integers k and l with 2 <= k <= n and 1 <= l <= kappa(k)(G), the (k, l)-rainbow index rx(k,l)(G) of G is the minimum number of colors needed in an edge coloring of G such that G contains at least l internally disjoint rainbow trees connecting S for every set S of k vertices of G. The numbers rx(k,l)(K-n) are determined for all possible values k and l when n <= 6. It is also shown that for l is an element of {1, 2}, rx(3,l)(K-n) = 3 for all n >= 6. (C) 2009 Wiley Periodicals, Inc. NETWORKS, Vol. 55(4), 360-367 2010
