5 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
An updated survey on rainbow connections of graphs - a dynamic survey
The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow -connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study
The Vertex-Rainbow Index of A Graph
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 introduce the concept of k-vertex-rainbow index rvxk(G) in this paper. In this paper, sharp upper and lower bounds of rvxk(G) are given for a connected graph G of order n, that is, 0 ≤ rvxk(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 rvxk(G) ≤ ℓ, where 2 ≤ ℓ ≤ n − 2 and 2 ≤ k ≤ n. Upper and lower bounds on t(n, k, ℓ) are also obtained
The Vertex-Rainbow Index of A Graph
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 introduce the concept of k-vertex-rainbow index rvxk(G) in this paper. In this paper, sharp upper and lower bounds of rvxk(G) are given for a connected graph G of order n, that is, 0 ≤ rvxk(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 rvxk(G) ≤ ℓ, where 2 ≤ ℓ ≤ n − 2 and 2 ≤ k ≤ n. Upper and lower bounds on t(n, k, ℓ) are also obtained