Rainbow connections in digraphs

Abstract

An arc-coloured path in a digraph is rainbow if its arcs have distinct colours. A vertex-coloured path is vertex rainbow if its internal vertices have distinct colours. A totally-coloured path is total rainbow if its arcs and internal vertices have distinct colours. An arc-coloured (resp. vertex-coloured, totally-coloured) digraph D is rainbow connected (resp. rainbow vertex-connected, total rainbow connected) if any two vertices of D are connected by a rainbow (resp. vertex rainbow, total rainbow) path. The rainbow connection number (resp. rainbow vertex-connection number, total rainbow connection number) of a digraph D is the smallest number of colours needed to make D rainbow connected (resp. rainbow vertex-connected, total rainbow connected).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.