Subdivisions of Oriented Cycles in Digraphs with Hamiltonian directed path

TL;DR

This paper improves bounds on the chromatic number needed in digraphs with Hamiltonian paths to guarantee subdivisions of all oriented cycles of a given length, advancing understanding of a longstanding conjecture.

Contribution

It reduces the chromatic bound from 3n to 2n for Hamiltonian digraphs and extends results to digraphs with Hamiltonian paths and high chromatic number.

Findings

Every 2n-chromatic Hamiltonian digraph contains a subdivision of every oriented cycle of order n.

Digraphs with a Hamiltonian directed path and chromatic number at least 12n-5 contain subdivisions of all cycles of length n.

The results deepen understanding of conditions needed for the conjecture involving subdivisions of oriented cycles.

Abstract

Cohen et al. conjectured that for every oriented cycle $C$ there exist an integer $f(C)$ such that every strong $f(C)$-chromatic digraph contains a subdivision of $C$. El Joubbeh confirmed this conjecture for Hamiltonian digraphs. Indeed, he showed that every $3n$-chromatic Hamiltonian digraph contains a subdivision of every oriented cycle of order $n$. In this article, we improve this bound to $2n$. Furthermore, we show that, if $D$ is a digraph containing a Hamiltonian directed path with chromatic number at least $12n-5$, then $D$ contains a subdivision of every oriented cycle of order $n$. Note that a digraph containing a Hamiltonian directed path need not be strongly connected. Thus, our current result provides a deeper understanding of the condition that may be needed to fully solve the conjecture.