Steiner (JGT 2024) proved that given a digraph F and pairs of integers \((r_e, q_e)\) with \(q_e\ge 2\) for every arc e of F, there is an integer N such that every digraph of dichromatic number at least N contains a subdivision of F in which for every arc e of F, the corresponding branching path has length \(r_e\pmod {q_e}\) . We extend this result to directed cycles containing a specified set of arcs. We show that given such F and \(\{(r_e, q_e)\}_{e\in A(F)}\) , there is an integer N such that if the minimum number of parts in a vertex partition \(\mathcal {P}\) of a digraph D such that each part has no directed cycles containing an arc of \(Z\subseteq A(D)\) is at least N, then D contains a subdivision of F in which for every arc e of F, the corresponding branching path has \(r_e\pmod {q_e}\) many arcs of Z. We prove our result in a slightly more general setting, by considering, given a digraph D and two sets \(Z_1\) and \(Z_2\) of arcs in D, the minimum number of parts in a vertex partition \(\mathcal {P}\) of D such that for every \(X\in \mathcal {P}\) , the subdigraph of D induced by X contains no directed cycle C with \(|A(C)\cap Z_1|\ne |A(C)\cap Z_2|\) . By setting \((Z_1, Z_2)=(Z, \emptyset )\) for \(Z\subseteq A(D)\) , we get an extension of the theorem of Steiner for directed cycles containing an arc of Z.
