The vertex-rainbow index of a graph
Abstract
The k-rainbow index rxk(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 introduced the concept of k-vertex-rainbow index rvxk(G) in this paper. For a graph G=(V,E) and a set S⊂eq V of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply, an S-tree) is a such subgraph T=(V',E') of G that is a tree with S⊂eq V'. For S⊂eq V(G) and |S|≥ 2, an S-Steiner tree T is said to be a vertex-rainbow S-tree if the vertices of V(T) S have distinct colors. For a fixed integer k with 2≤ k≤ n, the vertex-coloring c of G is called a k-vertex-rainbow coloring if for every k-subset S of V(G) there exists a vertex-rainbow S-tree. In this case, G is called vertex-rainbow k-tree-connected. The minimum number of colors that are needed in a k-vertex-rainbow coloring of G is called the k-vertex-rainbow index of G, denoted by rvxk(G). When k=2, rvx2(G) is nothing new but the vertex-rainbow connection number rvc(G) of G. In this paper, sharp upper and lower bounds of srvxk(G) are given for a connected graph G of order n,\ that is, 0≤ srvxk(G)≤ n-2. We obtain the Nordhaus-Guddum results for 3-vertex-rainbow index, and show that rvx3(G)+rvx3(G)=4 for n=4 and 2≤ rvx3(G)+rvx3(G)≤ n-1 for n≥ 5. Let t(n,k,) denote the minimal size of a connected graph G of order n with rvxk(G)≤ , where 2≤ ≤ n-2 and 2≤ k≤ n. The upper and lower bounds for t(n,k,) are also obtained.