The rainbow vertex-index of complementary graphs

Year 2015, Volume: 2 Issue: 3, 157 - 161, 14.09.2015

Fengnan Yanling Zhao Wang Chengfu Ye Shumin Zhang

https://doi.org/10.13069/jacodesmath.80607

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

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