84,119 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
On Rainbow Connection Number and Connectivity
Rainbow connection number, , of a connected graph is the minimum
number of colours needed to colour its edges, 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 investigate the relationship of rainbow connection number with
vertex and edge connectivity. It is already known that for a connected graph
with minimum degree , the rainbow connection number is upper bounded by
[Chandran et al., 2010]. This directly gives an upper
bound of and for rainbow
connection number where and , respectively, denote the edge
and vertex connectivity of the graph. We show that the above bound in terms of
edge connectivity is tight up-to additive constants and show that the bound in
terms of vertex connectivity can be improved to , for any . We conjecture that rainbow connection
number is upper bounded by and show that it is true for
. We also show that the conjecture is true for chordal graphs and
graphs of girth at least 7.Comment: 10 page
- …