New Hardness Results in Rainbow Connectivity

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph such that the graph is (strong) rainbow connected. It is known that for even k to decide whether the rainbow connectivity of a graph is at most k or not is NP-hard. It was conjectured that for all k, to decide whether rc(G) ≤ k is NP-hard. In this paper we prove this conjecture. We also show that it is NP-hard to decide whether src(G) ≤ k or not even when G is a