A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. 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 rainbow $u-v$ geodesic, then $G$ is strong rainbow vertex-connected. The minimum number $k$ for which there exists a $k$-vertex-coloring of $G$ that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of $G$, denoted by $srvc(G)$. Observe that $rvc(G)\leq srvc(G)$ for any nontrivial connected graph $G$. In this paper, sharp upper and lower bounds of $srvc(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq srvc(G)\leq n-2$. Graphs of order $n$ such that $srvc(G)= 1, 2, n-2$ are characterized, respectively. It is also shown that, for each pair $a, b$ of integers with $a\geq 5$ and $b\geq (7a-8)/5$, there exists a connected graph $G$ such that $rvc(G)=a$ and $srvc(G)=b$.
[1]
Raphael Yuster,et al.
On Rainbow Connection
,
2008,
Electron. J. Comb..
[2]
Xueliang Li,et al.
A solution to a conjecture on two rainbow connection numbers of a graph
,
2012,
Ars Comb..
[3]
Yongtang Shi,et al.
On the Rainbow Vertex-Connection
,
2013,
Discuss. Math. Graph Theory.
[4]
Raphael Yuster,et al.
Hardness and Algorithms for Rainbow Connectivity
,
2009,
STACS.
[5]
Xueliang Li,et al.
Rainbow Connections of Graphs: A Survey
,
2011,
Graphs Comb..
[6]
Raphael Yuster,et al.
The rainbow connection of a graph is (at most) reciprocal to its minimum degree
,
2010,
J. Graph Theory.
[7]
Xueliang Li,et al.
The complexity of determining the rainbow vertex-connection of a graph
,
2011,
Theor. Comput. Sci..
[8]
Garry L. Johns,et al.
Rainbow connection in graphs
,
2008
.
[9]
Xueliang Li,et al.
Rainbow Connections of Graphs
,
2012
.
[10]
J. A. Bondy,et al.
Graph Theory
,
2008,
Graduate Texts in Mathematics.