The Vertex-Rainbow Index of A Graph

The k-rainbow index rx_k(G) of a connected graph G was introduced by Chartrand, Okamoto and Zhang in 2010. As a natural counterpart of the k-rainbow index, we introduce the concept of k-vertex-rainbow index rvx_k(G) in this paper. In this paper, sharp upper and lower bounds of rvx_k(G) are given for a connected graph G of order n, that is, 0 ≤ rvx_k(G) ≤ n − 2. We obtain Nordhaus-Gaddum results for 3-vertex-rainbow index of a graph G of order n, and show that rvx₃(G) + rvx₃(Ḡ) = 4 for n = 4 and 2 ≤ rvx₃(G) + rvx₃(Ḡ) ≤ n − 1 for n ≥ 5. Let t(n, k, ℓ) denote the minimal size of a connected graph G of order n with rvx_k(G) ≤ ℓ, where 2 ≤ ℓ ≤ n − 2 and 2 ≤ k ≤ n. Upper and lower bounds on t(n, k, ℓ) are also obtained.