Subdivisions of Oriented Cycles in Digraphs with Hamiltonian directed path

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.