Dichromatic number of chordal graphs

TL;DR

This work investigates the dichromatic number $\vec{\chi}(D)$ of digraphs that are orientations or super-orientations of chordal graphs. It shows both limitations and tight possibilities: interval-graph orientations can force $\vec{\chi}$ to be as large as $\lceil k/2 \rceil$ when $\omega(G)=k$, while carefully constructed cograph orientations achieve $\vec{\chi}=\omega(G)$; it then provides several tight or near-tight bounds under constraints on the bidirected graph $B(D)$, including bounded maximum degree, bounded Mad, and $C_4$-free structures, with corresponding constructions that realize these bounds. The results illuminate how chordality and bidirected-graph structure govern dicolourings and suggest concrete open questions about optimal bounds and extensions to broader graph classes. Overall, the paper advances understanding of dichromatic colouring in chordal-related graph families and offers a framework for exploring how local bidirected constraints affect global 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.

Figures (5)

• Figure 1: The oriented interval graph $D_3$ (bits of $b_i^\ell$ are read from left to right).
• Figure 2: The oriented graphs $\Vec{G}_1$, $\Vec{G}_2$ and $\Vec{G}_3$.
• Figure 3: The digraphs $D_{1,0}$, $D_{3,0}$ and $D_{5,0}$.
• Figure 4: A chordal graph $G$ (on the left) and its valid tree-decomposition $T$ (on the right). The orange dashed arcs represent the chosen maximum path $P$. The ordering $a_1,\dots ,a_n$ of $V(G)$ we built is $a,b,c,i,j,h,d,l,e,f,g,k,m$.
• Figure 5: The digraph $D_{3,n}$.

Theorems & Definitions (32)

• Theorem 1: Andres and Hochstättler andresJGT79
• Theorem 2
• Theorem 3
• Proposition 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Corollary 8
• Proposition 9: Folklore