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