Subdivisions with congruence constraints in digraphs of large chromatic number

TL;DR

This paper proves that large dichromatic number in digraphs guarantees the existence of subdivisions of a fixed digraph with specified length congruence constraints on each subdivided arc, generalizing Thomassen's undirected graph result.

Contribution

It extends Thomassen's undirected graph subdivision result to directed graphs with congruence constraints, providing a new proof and broadening understanding of digraph structure.

Findings

Large dichromatic number ensures specific subdivided structures.

Generalization of Thomassen's undirected graph result to directed graphs.

Provides a concise proof technique for subdivision existence.

Abstract

We prove that for every digraph $F$ and every assignment of pairs of integers $(r_e,q_e)_{e \in A(F)}$ to its arcs there exists an integer $N$ such that every digraph $D$ with dichromatic number at least $N$ contains a subdivision of $F$ in which $e$ is subdivided into a directed path of length congruent to $r_e$ modulo $q_e$, for every $e \in A(F)$. This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result.