35 research outputs found
Rainbow Connection Number and Connected Dominating Sets
Rainbow connection number rc(G) of a connected graph G is the minimum number
of colours needed to colour the edges of G, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we show that for every connected graph G, with minimum degree at
least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2,
where {\gamma}_c(G) is the connected domination number of G. Bounds of the form
diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special
graph classes follow as easy corollaries from this result. This includes
interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and
chain graphs all with minimum degree at least 2 and connected. We also show
that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of
these cases, we also demonstrate the tightness of the bounds. An extension of
this idea to two-step dominating sets is used to show that for every connected
graph on n vertices with minimum degree {\delta}, the rainbow connection number
is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of
Schiermeyer (2009), improving the previously best known bound of 20n/{\delta}
by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up
to additive factors by a construction of Caro et al. (2008).Comment: 14 page
Rainbow connection in -connected graphs
An edge-colored graph is rainbow connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connection
number of a connected graph , denoted by , is the smallest number of
colors that are needed in order to make rainbow connected. In this paper,
we proved that for all -connected graphs.Comment: 7 page
Note on minimally -rainbow connected graphs
An edge-colored graph , where adjacent edges may have the same color, is
{\it rainbow connected} if every two vertices of are connected by a path
whose edge has distinct colors. A graph is {\it -rainbow connected} if
one can use colors to make rainbow connected. For integers and
let denote the minimum size (number of edges) in -rainbow connected
graphs of order . Schiermeyer got some exact values and upper bounds for
. However, he did not get a lower bound of for . In this paper, we improve his lower bound of
, and get a lower bound of for .Comment: 8 page
Rainbow connection number, bridges and radius
Let be a connected graph. The notion \emph{the rainbow connection number
} of a graph was introduced recently by Chartrand et al. Basavaraju
et al. showed that for every bridgeless graph with radius , , and the bound is tight. In this paper, we prove that if is a
connected graph, and is a connected -step dominating set of ,
then has a connected -step dominating set
such that , where is
the number of bridges in . Furthermore, for a connected
graph with radius , let be the center of , and .
Then has connected dominating sets
satisfying , and , where is the
number of bridges in . From the result, we
can get that if for all , then ; if for all , then
, the number of bridges of . This generalizes the
result of Basavaraju et al.Comment: 8 page