16,601 research outputs found
The 3-rainbow index of a graph
Let be a nontrivial connected graph with an edge-coloring , where adjacent edges may be
colored the same. A tree in is a if no two edges of
receive the same color. For a vertex subset , a tree that
connects in is called an -tree. The minimum number of colors that
are needed in an edge-coloring of such that there is a rainbow -tree for
each -subset of is called -rainbow index, denoted by
. In this paper, we first determine the graphs whose 3-rainbow index
equals 2, , , respectively. We also obtain the exact values of
for regular complete bipartite and multipartite graphs and wheel
graphs. Finally, we give a sharp upper bound for of 2-connected
graphs and 2-edge connected graphs, and graphs whose attains the
upper bound are characterized.Comment: 13 page
Graphs with 3-rainbow index and
Let be a nontrivial connected graph with an edge-coloring
, where adjacent edges
may be colored the same. A tree in is a if no two edges
of receive the same color. For a vertex set , the tree
connecting in is called an -tree. The minimum number of colors that
are needed in an edge-coloring of such that there is a rainbow -tree for
each -set of is called the -rainbow index of , denoted by
. In \cite{Zhang}, they got that the -rainbow index of a tree is
and the -rainbow index of a unicyclic graph is or . So
there is an intriguing problem: Characterize graphs with the -rainbow index
and . In this paper, we focus on , and characterize the graphs
whose 3-rainbow index is and , respectively.Comment: 14 page
Note on the upper bound of the rainbow index of a graph
A path in an edge-colored graph , where adjacent edges may be colored the
same, is a rainbow path if every two edges of it receive distinct colors. The
rainbow connection number of a connected graph , denoted by , is the
minimum number of colors that are needed to color the edges of such that
there exists a rainbow path connecting every two vertices of . Similarly, a
tree in is a rainbow~tree if no two edges of it receive the same color. The
minimum number of colors that are needed in an edge-coloring of such that
there is a rainbow tree connecting for each -subset of is
called the -rainbow index of , denoted by , where is an
integer such that . Chakraborty et al. got the following result:
For every , a connected graph with minimum degree at least
has bounded rainbow connection, where the bound depends only on
. Krivelevich and Yuster proved that if has vertices and the
minimum degree then . This bound was later
improved to by Chandran et al. Since , a
natural problem arises: for a general determining the true behavior of
as a function of the minimum degree . In this paper, we
give upper bounds of in terms of the minimum degree in
different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected
-step dominating sets, connected -dominating sets and -dominating
sets of .Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by
other author
The -rainbow index of random graphs
A tree in an edge colored graph is said to be a rainbow tree if no two edges
on the tree share the same color. Given two positive integers , with
, the \emph{-rainbow index} of is the
minimum number of colors needed in an edge-coloring of such that for any
set of vertices of , there exist internally disjoint rainbow
trees connecting . This concept was introduced by Chartrand et. al., and
there have been very few related results about it. In this paper, We establish
a sharp threshold function for and
respectively, where and are
the usually defined random graphs.Comment: 7 pages. arXiv admin note: substantial text overlap with
arXiv:1212.6845, arXiv:1310.278
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