Subdivisions in dicritical digraphs with large order or digirth

TL;DR

The paper advances directed-topological containment by establishing bounds and constructions that force subdivisions under high dichromatic number, high digirth, or large out-degree. It proves an improved bound for containing complete digraph subdivisions via $ ext{mader}_{\vec{χ}}$, and develops general mechanisms (via acyclic sets and digirth constraints) to guarantee subdivisions of a wide class of digraphs, including cycles, trees, and spindle structures like $C(k,k)$. It also shows both negative and positive phenomena for long directed paths in dicritical digraphs, and provides strong results for subdivisions in digraphs with large out-degree under digirth thresholds, including oriented graphs and out-stars. The results collectively extend Mader-type theory to the directed realm, offering concrete thresholds and constructions (such as $D_{k,n}$ and $C(k,k)$ substructures) that delineate when subdivisions are forced under structural constraints, while outlining key open problems about exact Mader-type numbers and their dependence on degree and girth.

Abstract

Aboulker et al. proved that a digraph with large enough dichromatic number contains any fixed digraph as a subdivision. The dichromatic number of a digraph is the smallest order of a partition of its vertex set into acyclic induced subdigraphs. A digraph is dicritical if the removal of any arc or vertex decreases its dichromatic number. In this paper we give sufficient conditions on a dicritical digraph of large order or large directed girth to contain a given digraph as a subdivision. In particular, we prove that (i) for every integers $k,\ell$, large enough dicritical digraphs with dichromatic number $k$ contain an orientation of a cycle with at least $\ell$ vertices; (ii) there are functions $f,g$ such that for every subdivision $F^*$ of a digraph $F$, digraphs with directed girth at least $f(F^*)$ and dichromatic number at least $g(F)$ contain a subdivision of $F^*$, and if $F$ is a tree, then $g(F)=|V(F)|$; (iii) there is a function $f$ such that for every subdivision $F^*$ of $TT_3$ (the transitive tournament on three vertices), digraphs with directed girth at least $f(F^*)$ and minimum out-degree at least $2$ contain $F^*$ as a subdivision.

Figures (5)

• Figure 1: The digraph $D_{10}$.
• Figure 2: The digraph $D_{3,7}$. The antidirected path $P$ is in blue.
• Figure 3: The structure of $C$ in $D$. The solid and dashed arcs represent the arcs of $C$. A dotted arc from $u$ to $v$ illustrates the existence of two internally-disjoint directed path from $u$ to $v$ in $D$.
• Figure 4: An example of a $P$-tricot of size $3$. Dotted arcs represent directed paths.
• Figure 5: An illustration of the paths $P_i'$ and $P_j'$.

Theorems & Definitions (49)

• Theorem 1: Aboulker et al. aboulkerEJC26
• Theorem 2
• Theorem 3
• Conjecture 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Lemma 9: Aboulker et al. aboulkerEJC26
• Lemma 10