Abstract
The
k
-rainbow index
r
x
k
(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 introduced the concept of
k
-vertex-rainbow index
rv
x
k
(G)
in this paper. For a graph
G=(V,E)
and a set
S⊆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⊆
V
′
. For
S⊆V(G)
and
|S|≥2
, an
S
-Steiner tree
T
is said to be a \emph{vertex-rainbow
S
-tree} if the vertices of
V(T)∖S
have distinct colors. For a fixed integer
k
with
2≤k≤n
, the vertex-coloring
c
of
G
is called a \emph{
k
-vertex-rainbow coloring} if for every
k
-subset
S
of
V(G)
there exists a vertex-rainbow
S
-tree. In this case,
G
is called \emph{vertex-rainbow
k
-tree-connected}. The minimum number of colors that are needed in a
k
-vertex-rainbow coloring of
G
is called the \emph{
k
-vertex-rainbow index} of
G
, denoted by
rv
x
k
(G)
. When
k=2
,
rv
x
2
(G)
is nothing new but the vertex-rainbow connection number
rvc(G)
of
G
. In this paper, sharp upper and lower bounds of
srv
x
k
(G)
are given for a connected graph
G
of order
n
,\ that is,
0≤srv
x
k
(G)≤n−2
. We obtain the Nordhaus-Guddum results for
3
-vertex-rainbow index, and show that
rv
x
3
(G)+rv
x
3
(
G
¯
¯
¯
¯
)=4
for
n=4
and
2≤rv
x
3
(G)+rv
x
3
(
G
¯
¯
¯
¯
)≤n−1
for
n≥5
. Let
t(n,k,ℓ)
denote the minimal size of a connected graph
G
of order
n
with
rv
x
k
(G)≤ℓ
, where
2≤ℓ≤n−2
and
2≤k≤n
. The upper and lower bounds for
t(n,k,ℓ)
are also obtained.