Subdivisions in digraphs of large out-degree or large dichromatic number

TL;DR

This paper investigates conditions under which large out-degree or high dichromatic number in digraphs guarantee the presence of subdivisions of certain graphs, advancing understanding of Mader's conjecture and related parameters.

Contribution

It proves that digraphs with large out-degree contain subdivisions of specific oriented paths and trees, and establishes bounds on dichromatic number ensuring subdivisions of all small digraphs.

Findings

Digraphs with sufficiently large out-degree contain subdivisions of certain oriented paths and trees.

High dichromatic number guarantees the presence of subdivisions of all small digraphs.

Provides bounds relating dichromatic number to the existence of subdivisions.

Abstract

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph $D$ is the smallest integer $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs. We show that any digraph with dichromatic number greater than $4^m (n-1)$ contains every digraph with $n$ vertices and $m$ arcs as a subdivision.