Paper Info
Title
Out-colourings of digraphs.
Abstract
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 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 (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+ck has such a partition. This is tight up to the value of c.
Year
DOI
Venue
2017
10.1002/jgt.22476
JOURNAL OF GRAPH THEORY
Keywords
DocType
Volume
degree constrained partitioning of digraphs,out-colourings,Paley tournament,polynomial algorithm,probabilistic method,semicomplete digraphs,tournaments
Journal
93.0
Issue
ISSN
Citations 
1.0
0364-9024
0
PageRank 
References 
Authors
0.34
0
3
Name
Order
Citations
PageRank
Noga Alon1104681688.16
Jørgen Bang-Jensen257368.96
Stéphane Bessy311719.68