4 research outputs found
The rainbow vertex-index of complementary graphs
A vertex-colored graph is \emph{rainbow vertex-connected} if two vertices are connected by a path whose internal vertices have distinct colors. The \emph{rainbow vertex-connection number} of a connected graph , denoted by , is the smallest number of colors that are needed in order to make rainbow vertex-connected. If for every pair of distinct vertices, contains a vertex-rainbow geodesic, then is \emph{strongly rainbow vertex-connected}. The minimum for which there exists a -coloring of that results in a strongly rainbow-vertex-connected graph is called the \emph{strong rainbow vertex number} of . Thus for every nontrivial connected graph . A tree in is called a \emph{rainbow vertex tree} if the internal vertices of receive different colors. For a graph and a set of at least two vertices, \emph{an -Steiner tree} or \emph{a Steiner tree connecting } (or simply, \emph{an -tree}) is a such subgraph of that is a tree with . For and , an -Steiner tree is said to be a \emph{rainbow vertex -tree} if the internal vertices of receive distinct colors. The minimum number of colors that are needed in a vertex-coloring of such that there is a rainbow vertex -tree for every -set of is called the {\it -rainbow vertex-index} of , denoted by . In this paper, we first investigate the strong rainbow vertex-connection of complementary graphs. The -rainbow vertex-index of complementary graphs are also studied
The rainbow vertex-index of complementary graphs
A vertex-colored graph is \emph{rainbow vertex-connected} if two vertices are connected by a path whose internal vertices have distinct colors. The \emph{rainbow vertex-connection number} of a connected graph , denoted by , is the smallest number of colors that are needed in order to make rainbow vertex-connected. If for every pair of distinct vertices, contains a vertex-rainbow geodesic, then is \emph{strongly rainbow vertex-connected}. The minimum for which there exists a -coloring of that results in a strongly rainbow-vertex-connected graph is called the \emph{strong rainbow vertex number} of . Thus for every nontrivial connected graph . A tree in is called a \emph{rainbow vertex tree} if the internal vertices of receive different colors. For a graph and a set of at least two vertices, \emph{an -Steiner tree} or \emph{a Steiner tree connecting } (or simply, \emph{an -tree}) is a such subgraph of that is a tree with . For and , an -Steiner tree is said to be a \emph{rainbow vertex -tree} if the internal vertices of receive distinct colors. The minimum number of colors that are needed in a vertex-coloring of such that there is a rainbow vertex -tree for every -set of is called the {\it -rainbow vertex-index} of , denoted by . In this paper, we first investigate the strong rainbow vertex-connection of complementary graphs. The -rainbow vertex-index of complementary graphs are also studied