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

TL;DR

This paper introduces a new variant of the dichromatic number for digraphs with prescribed arc sets and proves that large enough values guarantee the existence of certain subdivided subdigraphs satisfying modular arc constraints.

Contribution

It generalizes previous results by establishing conditions under which large dichromatic-like parameters ensure specific subdivided structures with modular arc constraints.

Findings

Large $mbda$ implies existence of subdivided subdigraphs with modular arc properties.

Generalizes Steiner's theorem to broader modular and arc set conditions.

Provides bounds for the parameter $mbda$ ensuring the substructure existence.

Abstract

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 $(a_e, b_e, Z_1, Z_2)=(1, 1, A(D), \emptyset)$.