196 research outputs found
On strong rainbow connection number
A path in an edge-colored graph, where adjacent edges may be colored the
same, is a rainbow path if no two edges of it are colored the same. For any two
vertices and of , a rainbow geodesic in is a rainbow
path of length , where is the distance between and .
The graph is strongly rainbow connected if there exists a rainbow
geodesic for any two vertices and in . The strong rainbow connection
number of , denoted , is the minimum number of colors that are
needed in order to make strong rainbow connected. In this paper, we first
investigate the graphs with large strong rainbow connection numbers. Chartrand
et al. obtained that is a tree if and only if , we will show that
, so is not a tree if and only if , where
is the number of edge of . Furthermore, we characterize the graphs
with . We next give a sharp upper bound for according to
the number of edge-disjoint triangles in graph , and give a necessary and
sufficient condition for the equality.Comment: 16 page
The generalized 3-connectivity of Cartesian product graphs
The generalized connectivity of a graph, which was introduced recently by
Chartrand et al., is a generalization of the concept of vertex connectivity.
Let be a nonempty set of vertices of , a collection
of trees in is said to be internally disjoint trees
connecting if and for
any pair of distinct integers , where . For an integer
with , the -connectivity of is the
greatest positive integer for which contains at least internally
disjoint trees connecting for any set of vertices of .
Obviously, is the connectivity of . Sabidussi showed
that for any two connected graphs
and . In this paper, we first study the 3-connectivity of the Cartesian
product of a graph and a tree , and show that if
, then ;
if , then .
Furthermore, for any two connected graphs and with
, if , then ; if , then
. Our result could be seen as
a generalization of Sabidussi's result. Moreover, all the bounds are sharp.Comment: 17 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
A study of dynamical processes in the Orion KL region using ALMA-- Probing molecular outflow and inflow
This work reports a high spatial resolution observations toward Orion KL
region with high critical density lines of CHCN (12-11) and
CHOH (8-7) as well as continuum at 1.3 mm band.
The observations were made using the Atacama Large Millimeter/Submillimeter
Array with a spatial resolution of 1.5 and sensitives
about 0.07 K and 0.18 K for continuum and line, respectively. The
observational results showed that the gas in the Orion KL region consists of
jet-propelled cores at the ridge and dense cores at east and south of the
region, shaped like a wedge ring. The outflow has multiple lobes, which may
originate from an explosive ejection and is not driven by young stellar
objects. Four infrared bubbles were found in the Spitzer/IRAC emissions. These
bubbles, the distributions of the previously found H jets, the young
stellar objects and molecular gas suggested that BN is the explosive center.
The burst time was estimated to be 1300 years. In the mean time,
signatures of gravitational collapse toward Source I and hot core were detected
with material infall velocities of 1.5 km~s and 0.6 km~s,
corresponding to mass accretion rates of 1.210M_{\sun}/Yr and
8.010M_{\sun}/Yr, respectively. These observations may support
that high-mass stars form via accretion model, like their low-mass
counterparts.Comment: Accepted to Ap
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