Decomposition of oriented k-partition-connected digraphs

Definitions and mathematical statement

Let $D=(V,A)$ be a finite loopless digraph, with a vertex $r$ of indegree zero, and let $k\ge1$. Distinct parallel arcs remain distinct edges when directions are forgotten. Say that the underlying multigraph is $k$-partition-connected when every partition $\mathcal P$ of $V$ into nonempty classes satisfies \[ |\{a\in A:\text{the endpoints of }a\text{ lie in different classes}\}| \ge k(|\mathcal P|-1). \] Assume this condition and $d_D^-(v)\ge k$ for each $v\ne r$.

Must $A$ have a partition $A_1\sqcup\cdots\sqcup A_k$ such that every spanning subdigraph $(V,A_i)$ is weakly connected and \[ d^-_{(V,A_i)}(v)\ge1 \quad(v\ne r,\ 1\le i\le k)? \] Weak connectivity means connectivity after discarding arc directions. The vertex set is retained in every part, including vertices that would otherwise be isolated.

The conclusion is not a decomposition into rooted arborescences: directed cycles are permitted and reachability from $r$ is not separately required. The incoming-degree and undirected connectivity conditions must nevertheless hold in the same parts. Splitting the arcs once to satisfy degrees and a second time to obtain connectivity would not establish a common decomposition satisfying both requirements.

Short English statement

Do the natural partition-connectivity and indegree conditions guarantee a decomposition into weakly connected spanning digraphs with positive indegree away from the root?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000028 (AMR-029-0028), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Egres Open - List of open problems, Open-problem page: Decomposition of oriented k-partition-connected digraphs. Original source indexed by AMR-029-0028. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
• S3 EGRES Open, Decomposition of oriented k-partition-connected digraphs. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Decomposition_of_oriented_k-partition-connected_digraphs.