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Dichromatic Number and Cycle Inversions

Pierre Charbit, Stéphan Thomassé

TL;DR

Addresses the problem of determining the dichromatic number $\\overrightarrow{\\chi}(D)$ of a digraph and relates it to vertex orderings and cycle structure. The approach establishes a three-way (incidence-matrix, arc-weighting, Farkas lemma) equivalence that yields a constructive criterion: $D$ admits a $k$-acyclic colouring iff every directed circuit $C$ has at least $|C|/k$ forward arcs with respect to some ordering $\\sigma$. The key contribution is a succinct, verifiable proof and a concrete method to partition $V(D)$ into $k$ acyclic colour classes via subgraphs $D_t$ created from a vector $z$ satisfying $^t z M\\le p$, with $p(a)=\\mathrm{forw}_{\\sigma}(a)-1/k$. A corollary shows one can reverse circuit orientations to force $\\overrightarrow{\\chi}(D')\\le 2$; this cycle-inversion approach extends to certain infinite digraphs (Ellis/Soukoup, 2020), highlighting the practical impact of the results.

Abstract

The results of this note were stated in the first author PhD manuscript in 2006 but never published. The writing of a proof given there was slightly careless and the proof itself scattered across the document, the goal of this note is to give a short and clear proof using Farkas Lemma. The first result is a characterization of the acyclic chromatic number of a digraph in terms of cyclic ordering. Using this theorem we prove that for any digraph, one can sequentially reverse the orientations of the arcs of a family of directed cycles so that the resulting digraph has acyclic chromatic number at most 2.

Dichromatic Number and Cycle Inversions

TL;DR

Addresses the problem of determining the dichromatic number of a digraph and relates it to vertex orderings and cycle structure. The approach establishes a three-way (incidence-matrix, arc-weighting, Farkas lemma) equivalence that yields a constructive criterion: admits a -acyclic colouring iff every directed circuit has at least forward arcs with respect to some ordering . The key contribution is a succinct, verifiable proof and a concrete method to partition into acyclic colour classes via subgraphs created from a vector satisfying , with . A corollary shows one can reverse circuit orientations to force ; this cycle-inversion approach extends to certain infinite digraphs (Ellis/Soukoup, 2020), highlighting the practical impact of the results.

Abstract

The results of this note were stated in the first author PhD manuscript in 2006 but never published. The writing of a proof given there was slightly careless and the proof itself scattered across the document, the goal of this note is to give a short and clear proof using Farkas Lemma. The first result is a characterization of the acyclic chromatic number of a digraph in terms of cyclic ordering. Using this theorem we prove that for any digraph, one can sequentially reverse the orientations of the arcs of a family of directed cycles so that the resulting digraph has acyclic chromatic number at most 2.
Paper Structure (3 sections, 3 theorems, 2 equations)

This paper contains 3 sections, 3 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. The main result and its proof
  3. A corollary about cycle inversions

Key Result

Theorem 2.1

A digraph $D$ can be partitioned into $k$ acyclic subgraphs (i.e. $\mathop{\mathrm{\overrightarrow{\chi}}}\nolimits(D)\leq k$) if and only if there is an ordering $\sigma$ of the vertices of $D$ such that for any circuit $C$, the number of arcs of $C$ going forward in $\sigma$ is at least $|C|/k$.

Theorems & Definitions (5)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • proof : of Theorem \ref{['thm:main']}
  • Corollary 3.1