Rainbow trees in graphs and generalized connectivity

Abstract

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 ⊆ 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 {T1,T2,…,Tℓ} of trees in G is said to be internally disjoint connecting S if these trees are pairwise edge-disjoint and V(Ti) ∩ V(Tj) = 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 κk(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 κk(Kn)=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 ≤ κk(G), the (k,ℓ)-rainbow index rxk,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 rxk,ℓ(Kn) are determined for all possible values k and ℓ when n ≤ 6. It is also shown that for ℓ ϵ {1, 2}, rx3,ℓ(Kn) = 3 for all n ≥ 6. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010