Out-colourings of Digraphs
Noga Alon, Joergen Bang-Jensen, St\'ephane Bessy

TL;DR
This paper investigates out-colourings of digraphs, establishing NP-completeness for 2-out-colourings, characterizing when such colourings exist in tournaments and semicomplete digraphs, and exploring partitions with out-degree constraints using probabilistic methods.
Contribution
It proves existence results for out-colourings in tournaments and semicomplete digraphs, including conditions for 2-out-colourings and balanced colourings, and introduces probabilistic bounds for vertex partitions with out-degree constraints.
Findings
Deciding 2-out-colouring is NP-complete.
Almost all strong semicomplete digraphs with minimum out-degree ≥3 have a 2-out-colouring, except P7.
Existence of partitions with out-degree constraints in semicomplete digraphs proven using probabilistic methods.
Abstract
We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an {\bf 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 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 {\bf balanced} such colouring, that is, the difference between the number of vertices that receive colour 1 and colour 2 is at…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Advanced Topology and Set Theory
