2,783 research outputs found
Hardness and Algorithms for Rainbow Connectivity
An edge-colored graph G is rainbow connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connectivity
of a connected graph G, denoted rc(G), is the smallest number of colors that
are needed in order to make G rainbow connected. In addition to being a natural
combinatorial problem, the rainbow connectivity problem is motivated by
applications in cellular networks. In this paper we give the first proof that
computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to
decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given
edge-colored (with an unbounded number of colors) graph is rainbow connected.
On the positive side, we prove that for every > 0, a connected graph
with minimum degree at least has bounded rainbow connectivity,
where the bound depends only on , and the corresponding coloring can
be constructed in polynomial time. Additional non-trivial upper bounds, as well
as open problems and conjectures are also pre sented
Rainbow -connectivity of random bipartite graphs
A path in an edge-colored graph is called a rainbow path if no two edges
of the path are colored the same. The minimum number of colors required to
color the edges of such that every pair of vertices are connected by at
least internally vertex-disjoint rainbow paths is called the rainbow
-connectivity of the graph , denoted by . For the random graph
, He and Liang got a sharp threshold function for the property
. In this paper, we extend this result to the case of
random bipartite graph .Comment: 15 pages. arXiv admin note: text overlap with arXiv:1012.1942 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
Rainbow Connection of Random Regular Graphs
An edge colored graph is rainbow edge connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connection of
a connected graph , denoted by , is the smallest number of colors
that are needed in order to make rainbow connected.
In this work we study the rainbow connection of the random -regular graph
of order , where is a constant. We prove that with
probability tending to one as goes to infinity the rainbow connection of
satisfies , which is best possible up to a hidden
constant
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