Rainbow connections in digraphs

In this paper, we study the rainbow connection, rainbow vertex-connection and total rainbow connection numbers of digraphs. We give some properties of these parameters and establish relations between them. The rainbow connection number and the rainbow vertex-connection number of a digraph D are both upper bounded by the order of D, while its total rainbow connection number is upper bounded by twice of its order. In particular, we prove that a digraph of order n has rainbow connection number n if and only if it is Hamiltonian and has three vertices with eccentricity n−1, that it has rainbow vertex-connection number n if and only if it has a Hamiltonian cycle C and three vertices with eccentricity n−1 such that no two of them are consecutive on C, and that it has total rainbow connection number 2n if and only if it has rainbow vertex-connection number n.