Dichromatic number of chordal graphs

TL;DR

This paper investigates the bounds on the dichromatic number of super-orientations of chordal graphs, providing new theoretical insights and specific examples related to cographs.

Contribution

It establishes bounds on the dichromatic number for super-orientations of chordal graphs and presents a family of orientations of cographs where the dichromatic number equals the clique number.

Findings

Bounds on dichromatic number for super-orientations of chordal graphs

A family of cograph orientations with dichromatic number equal to clique number

Theoretical framework connecting graph orientations and acyclic partitions

Abstract

The dichromatic number of a digraph is the minimum integer $k$ such that it admits a $k$-dicolouring, i.e. a partition of its vertices into $k$ acyclic subdigraphs. We say that a digraph $D$ is a super-orientation of an undirected graph $G$ if $G$ is the underlying graph of $D$. If $D$ does not contain any pair of symmetric arcs, we just say that $D$ is an orientation of $G$. In this work, we give both lower and upper bounds on the dichromatic number of super-orientations of chordal graphs. We also show a family of orientations of cographs for which the dichromatic number is equal to the clique number of the underlying graph.