The rainbow vertex-index of complementary graphs ∗

A vertex-colored graph G is rainbow vertex-connected if two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. If for every pair u, v of distinct vertices, G contains a vertex-rainbow u−v geodesic, then G is strongly rainbow vertex-connected. The minimum k for which there exists a k-coloring of G that results in a strongly rainbow-vertex-connected graph is called the strong rainbow vertex number srvc(G) of G. Thus rvc(G) ≤ srvc(G) for every nontrivial connected graph G. A tree T in G is called a rainbow vertex tree if the internal vertices of T receive different colors. For a graph G = (V,E) and a set S ⊆ 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 ⊆ V ′. For S ⊆ V (G) and |S| ≥ 2, an S-Steiner tree T is said to be a rainbow vertex S-tree if the internal vertices of T receive distinct colors. The minimum number of colors that are needed in a vertex-coloring of G such that there is a rainbow vertex S-tree for every k-set S of V (G) is called the k-rainbow vertex-index of G, denoted by rvxk(G). In this paper, we first investigate the strong rainbow vertex-connection of complementary graphs. The k-rainbow vertex-index of complementary graphs are also studied. 2010 MSC: 05C15, 05C40


Introduction
The graphs considered in this paper are finite undirected and simple graphs.We follow the notation of Bondy and Murty [1], unless otherwise stated.For a graph G, let V (G), E(G), n(G), m(G), and G, respectively, be the set of vertices, the set of edges, the order, the size, and the complement graph of G.
Let G be a nontrivial connected graph on which an edge-coloring c : E(G) → {1, 2, • • • , n}, n ∈ N, is defined, where adjacent edges may be colored the same.A path is rainbow if no two edges of it are colored the same.An edge-coloring graph G is rainbow connected if any two vertices are connected by a rainbow path.Clearly, if a graph is rainbow connected, it must be connected, whereas any connected graph has a trivial edge-coloring that makes it rainbow connected; just color each edge with a distinct color.Thus, in [4] L. Chen, X. Li, H. Lian defined the rainbow connection number of a connected graph G, denoted by rc(G), as the smallest number of colors that are needed in order to make G rainbow connected.They showed that rc(G) ≥ diam(G) where diam(G) denotes the diameter of G.For more results on the rainbow connection, we refer to the survey paper [2], [3], [4] and [12], and a new book [10] of Li and Sun.
In [8], Krivelevich and Yuster proposed the concept of rainbow vertex-connection.A vertex-colored graph G is rainbow vertex-connected if two vertices are connected by a path whose internal vertices have distinct colors.The rainbow vertex-connection number of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected.For more results on the rainbow vertex-connection, we refer to the survey paper [5] and [9].An easy observation is If for every pair u, v of distinct vertices, G contains a vertex-rainbow u − v geodesic, then G is strong rainbow vertex-connected.The definition of strongly rainbow vertex-connected was defined by Li et al. in [11].The minimum k for which there exists a k-coloring of G that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number srvc(G) of G. Thus rc(G) ≤ srvc(G) for every nontrivial connected graph G.
where s denote the number of pendent vertices in G.A tree T in G is called a rainbow vertex tree if the internal vertices of T receive different colors.For a graph G = (V, E) and a set S ⊆ 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 ⊆ V .For more problems on S-Steiner tree, we refer to [6] and [7].
For S ⊆ V (G) and |S| ≥ 2, an S-Steiner tree T is said to be a rainbow vertex S-tree if the internal vertices of T receive distinct colors.The minimum number of colors that are needed in an vertex-coloring of G such that there is a rainbow vertex S-tree for every k-set S of V (G) is called the k-rainbow vertexindex of G, denoted by rvx k (G).The vertex-rainbow index of a graph was first defined by Yaping Mao in [13].

The strong rainbow vertex-connection of complementary graphs
In this section, we investigate the rainbow vertex-connection number of a graph G according to some constraints to its complement G.We give some conditions to guarantee that srvc(G) is bounded by a constant.
We investigate the rainbow vertex-connection number of connected complement graphs of graphs with diameter at least 3.
Proof.We choose a vertex x with ecc For example, see Figure 1, a graph with diam(G) = 5.First of all, we see that G must be connected, since otherwise, diam(G) ≤ 2, contradicting the condition diam(G) ≥ 3.
In this case, k 1 , k 2 ≥ 3. We will show that diam( Ḡ) ≤ 2 in this case.For u, v ∈ V (G), we consider the following cases:

Proposition 1 . 1 .
Let G be a nontrivial connected graph of order n.Then (a) srvc(G) = 0 if and only if G is a complete graph; (b) srvc(G) = 1 if and only if diam(G) = 2 if and only if rvc(G) = 1.

Figure 1 .
Figure 1.Graphs for the proof of Theorem 2.
The same is true for u, v ∈ B. Subcase 1.2.u ∈ A and v ∈ B.If u = x, v ∈ B, then u is adjacent to all vertices in G[B] \ N 1 G (x).So d G (u, v) = 1 for v ∈ G[B] \ N 1 G (x).For v ∈ N 1 G (x), let P = ux 3 v, where x 3 ∈ N 3 G (x).Clearly, d G (u, v) = 2.If u = x, without loss of generality, we assume that u ∈ N 2 G (x) and v ∈ N 1 G (x).Let Q = ux 5 v, where x 5 ∈ N 5 G (x). Thus d G (u, v) = 2.From the above, we conclude that diam(G) ≤ 2. So, by Proposition 1(b), we have srvc(G) = 1.