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Orientations of cycles in digraphs of high chromatic number and high minimum out-degree

Hidde Koerts, Benjamin Moore, Sophie Spirkl

TL;DR

This work characterizes exactly which orientations $C$ of a cycle must appear in every digraph $D$ with no loops or parallel arcs, given large chromatic number and a linear lower bound on the minimum out-degree. The authors prove a complete dichotomy: $C$ is forced if and only if it has at least three blocks or consists of two blocks of length at least two, with in-degree duality and Thomassen-type reductions linking to the undirected case. The proof combines a directed analogue of Thomassen’s approach with Burr’s subtrees, cohesive-set techniques, and a careful two-case analysis, while also presenting explicit constructions (via directed shift graphs and blow-ups) showing that the two exceptional orientations—directed cycles and single-arc flips—are avoidable. Together, these results advance the understanding of directed chromatic thresholds for cycle orientations and connect to broader questions about fixed digraphs in dense high-chromatic digraphs.

Abstract

We characterize all orientations of cycles $C$ for which for every fixed $\varepsilon > 0$ there exists a constant $c \geq 1$ such that every digraph $D$ without loops or parallel arcs with $χ(D) \geq c$ and minimum out-degree at least $\varepsilon |V(D)|$ contains $C$ as a subdigraph. This generalizes a result of Thomassen.

Orientations of cycles in digraphs of high chromatic number and high minimum out-degree

TL;DR

This work characterizes exactly which orientations of a cycle must appear in every digraph with no loops or parallel arcs, given large chromatic number and a linear lower bound on the minimum out-degree. The authors prove a complete dichotomy: is forced if and only if it has at least three blocks or consists of two blocks of length at least two, with in-degree duality and Thomassen-type reductions linking to the undirected case. The proof combines a directed analogue of Thomassen’s approach with Burr’s subtrees, cohesive-set techniques, and a careful two-case analysis, while also presenting explicit constructions (via directed shift graphs and blow-ups) showing that the two exceptional orientations—directed cycles and single-arc flips—are avoidable. Together, these results advance the understanding of directed chromatic thresholds for cycle orientations and connect to broader questions about fixed digraphs in dense high-chromatic digraphs.

Abstract

We characterize all orientations of cycles for which for every fixed there exists a constant such that every digraph without loops or parallel arcs with and minimum out-degree at least contains as a subdigraph. This generalizes a result of Thomassen.

Paper Structure

This paper contains 5 sections, 10 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Standard notation and terminology
  3. Orientations of cycles containing $\rightarrow\leftarrow\rightarrow\leftarrow$ or $\rightarrow\rightarrow\leftarrow\leftarrow$
  4. Constructions for directed cycles and orientations of cycles with a single arc flipped
  5. Related problems

Key Result

Theorem 1.3

Let $\varepsilon > 0$ and $k \geq 2$ be fixed. For $C$ an orientation of a $k$-cycle, there exists a constant $c \geq 1$ such that every digraph $D$ without loops or parallel arcs with $\chi(D) \geq c$ and minimum out-degree at least $\varepsilon|V(D)|$ contains $C$ as a subdigraph if and only if $C

Figures (2)

  • Figure 1: Directed shift graph $\vec{S}_{5,3}$.
  • Figure 2: Illustration of the construction used in the proof of \ref{['thm:counterexample-flipped']}.

Theorems & Definitions (23)

  • Theorem 1.3
  • Theorem 1.4: burr1980subtrees
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 13 more