An edge-colored graph G is rainbow connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connectivity
of a connected graph G, denoted rc(G), is the smallest number of colors that
are needed in order to make G rainbow connected. In addition to being a natural
combinatorial problem, the rainbow connectivity problem is motivated by
applications in cellular networks. In this paper we give the first proof that
computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to
decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given
edge-colored (with an unbounded number of colors) graph is rainbow connected.
On the positive side, we prove that for every ϵ > 0, a connected graph
with minimum degree at least ϵn has bounded rainbow connectivity,
where the bound depends only on ϵ, and the corresponding coloring can
be constructed in polynomial time. Additional non-trivial upper bounds, as well
as open problems and conjectures are also pre sented