The rainbow vertex-index of complementary graphs

Abstract

A vertex-colored graph $G$ 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 $G$, 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 \emph{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 \emph{strong rainbow
vertex number} $srvc(G)$ of $G$. Thus $rvc(G)\leq srvc(G)$ for every
nontrivial connected graph $G$. A tree $T$ in $G$ is called a
\emph{rainbow vertex tree} if the internal vertices of $T$ receive
different colors. For a graph $G=(V,E)$ and a set $S\subseteq V$ of
at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner
tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such
subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For
$S\subseteq V(G)$ and $|S|\geq 2$, an $S$-Steiner tree $T$ is said
to be a \emph{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 {\it
$k$-rainbow vertex-index} of $G$, denoted by $rvx_k(G)$. 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.

Keywords

• Strong rainbow vertex-connection number, Complementary graph, Rainbow vertex S-tree, k-rainbowvertex-index

References

1. J. A. Bondy and U. S. R. Murty, Graph theory, GTM 244, Springer, 2008.
2. G. Chartrand, G. L. Johns, K. A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem., 133, 85–98, 2008.
3. G. Chartrand, G. L. Johns, K. A. McKeon and P. Zhang, The rainbow connectivity of a graph, Networks, 54(2), 75–81, 2009.
4. L. Chen, X. Li and H. Lian, Nordhaus-Gaddum-type theorem for rainbow connection number of graphs, Graphs Combin., 29(5), 1235–1247, 2013.
5. L. Chen, X. Li and M. Liu, Nordhaus-Gaddum-type theorem for the rainbow vertex connection number of a graph, Utilitas Math., 86, 335–340, 2011.
6. X. Cheng and D. Du, Steiner trees in industry, Kluwer Academic Publisher, Dordrecht, 2001.
7. D. Du and X. Hu, Steiner tree problems in computer communication networks, World Scientific, 2008.
8. M. Krivelevich and R. Yuster, The rainbow connection of a graph is (at most) reciprocal to its minimum degree three, J. Graph Theory, 63(3), 185-191, 2010.

9. X. Li and Y. Shi, On the rainbow vertex-connection, Discuss. Math. Graph Theory, 33(2), 307–313, 20 X. Li and Y. Sun, Rainbow connections of graphs, SpringerBriefs in Math., Springer, New York, 20 X. Li, Y. Mao and Y. Shi, The strong rainbow vertex-connection of graphs, Utilitas Math., 93, 213– 223, 2014.
10. X. Li, Y. Shi and Y. Sun, Rainbow connections of graphs–A survey, Graphs Combin., 29(1), 1-38, 20 Y. Mao, The vertex-rainbow index of a graph, arXiv:1502.00151v1 [math.CO], 31 Jan 2015.
11. E. A. Nordhaus and J. W. Gaddum, On complementary graphs, Amer. Math. Monthly, 63, 175–177, 19