Critical digraphs with few vertices

TL;DR

This paper proves that small k-dichromatic vertex-critical digraphs have disconnected complements, extending classical results from undirected graphs and answering a question posed by Bang-Jensen et al.

Contribution

It establishes a new property of small k-dichromatic vertex-critical digraphs, generalizing Gallai's theorem to directed graphs.

Findings

Every such digraph on at most 2k-2 vertices has a disconnected complement.

Answers a question of Bang-Jensen et al.

Generalizes Gallai's theorem to directed graphs.

Abstract

We show that every k-dichromatic vertex-critical digraph on at most 2k-2 vertices has a disconnected complement. This answers a question of Bang-Jensen et al., and generalises a classical theorem of Gallai on undirected vertex-critical graphs.