Rainbow connection number, rc(G), of a connected graph G is the minimum
number of colors needed to color its edges so that every pair of vertices is
connected by at least one path in which no two edges are colored the same (Note
that the coloring need not be proper). In this paper we study the rainbow
connection number with respect to three important graph product operations
(namely cartesian product, lexicographic product and strong product) and the
operation of taking the power of a graph. In this direction, we show that if G
is a graph obtained by applying any of the operations mentioned above on
non-trivial graphs, then rc(G) <= 2r(G)+c, where r(G) denotes the radius of G
and c \in {0,1,2}. In general the rainbow connection number of a bridgeless
graph can be as high as the square of its radius [Basavaraju et. al, 2010].
This is an attempt to identify some graph classes which have rainbow connection
number very close to the obvious lower bound of diameter (and thus the radius).
The bounds reported are tight upto additive constants. The proofs are
constructive and hence yield polynomial time (2 + 2/r(G))-factor approximation
algorithms.