On a variant of dichromatic number for digraphs with prescribed sets of arcs

In this paper, we consider a variant of dichromatic number on digraphs with prescribed sets of arcs. Let $D$ be a digraph and let $Z_1, Z_2$ be two sets of arcs in $D$. For a subdigraph $H$ of $D$, let $A(H)$ denote the set of all arcs of $H$. Let $\mu(D, Z_1, Z_2)$ be 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 $|A(C)\cap Z_1|\neq |A(C)\cap Z_2|$. For $Z_1=A(D)$ and $Z_2=\emptyset$, $\mu(D, Z_1, Z_2)$ is equal to the dichromatic number of $D$. We prove that for every digraph $F$ and every tuple $(a_e,b_e,r_e, q_e)$ of integers with $q_e\ge 2$ and $\gcd(a_e,q_e)=\gcd(b_e,q_e)=1$ for each arc $e$ of $F$, there exists an integer $N$ such that if $\mu(D, Z_1, Z_2)\ge N$, then $D$ contains a subdigraph isomorphic to a subdivision of $F$ in which each arc $e$ of $F$ is subdivided into a directed path~$P_e$ such that~$a_e|A(P_e)\cap Z_1|+b_e|A(P_e)\cap Z_2|\equiv {r_e}\pmod {q_e}$. This generalizes a theorem of Steiner [Subdivisions with congruence constraints in digraphs of large chromatic number, arXiv:2208.06358] which corresponds to the case when $(a_e, b_e, Z_1, Z_2)=(1, 1, A(D), \emptyset)$.