Rainbow connection number and independence number of a graph
Abstract
Let
G
be an edge-colored connected graph. A path of
G
is called rainbow if its every edge is colored by a distinct color.
G
is called rainbow connected if there exists a rainbow path between every two vertices of
G
. The minimum number of colors that are needed to make
G
rainbow connected is called the rainbow connection number of
G
, denoted by
rc(G)
. In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if
G
is a connected graph, then
rc(G)≤2α(G)−1
. Two examples
G
are given to show that the upper bound
2α(G)−1
is equal to the diameter of
G
, and therefore the best possible since the diameter is a lower bound of
rc(G)
.