Out-colourings of Digraphs

We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a 2-out-colouring) is NP-complete. We show that for every choice of positive integers r, k there exists a k-strong bipartite tournament which needs at least r colours in every out-colouring. Our main results are on tournaments and semicomplete digraphs. We prove that, except for the Paley tournament P7, every strong semicomplete digraph of minimum out-degree at least 3 has a 2-out-colouring. Furthermore, we show that every semicomplete digraph on at least 7 vertices has a 2-out-colouring if and only if it has a balanced such colouring, that is, the difference between the number of vertices that receive colour 1 and colour 2 is at most one. In the second half of the paper we consider the generalization of 2-out-colourings to vertex partitions (V1, V2) of a digraph D so that each of the three digraphs induced by respectively, the vertices of V1, the vertices of V2 and all arcs between V1 and V2 have minimum out-degree k for a prescribed integer k ≥ 1. Using probabilistic arguments we prove that there exists an absolute positive constant c so that every semicomplete digraph of minimum out-degree at least 2k + c √ k has such a partition. This is tight up to the value of c.