Out‐colourings of digraphs

Abstract

AbstractWe 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 ‐complete. We show that for every choice of positive integers there exists a ‐strong bipartite tournament, which needs at least colours in every out‐colouring. Our main results are on tournaments and semicomplete digraphs. We prove that, except for the Paley tournament , every strong semicomplete digraph of minimum out‐degree at least three has a 2‐out‐colouring. Furthermore, we show that every semicomplete digraph on at least seven 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 of a digraph so that each of the three digraphs induced by respectively, the vertices of , the vertices of and all arcs between and , have minimum out‐degree for a prescribed integer . Using probabilistic arguments, we prove that there exists an absolute positive constant so that every semicomplete digraph of minimum out‐degree at least has such a partition. This is tight up to the value of .