# Decomposition of oriented k-partition-connected digraphs

Let D=(V,A) be a digraph whose underlying graph is k-partition-connected, and let $r_0 \in V$ be a node of in-degree 0. Suppose that the in-degree of every other node is at least k. Is it true that D can be decomposed into k weakly connected spanning subgraphs, so that every node $v \in V-r_0$ has positive in-degree in each subgraph?

## Remarks

Let $M_1$ be the cycle matroid of the underlying undirected graph of D, and let $M_2$ be the partition matroid where each class consists of the set of edges entering a given node $v \in V-r_0$. The problem can be formulated as follows:

Is it true that if both $M_1$ and $M_2$ contain k disjoint spanning sets, then there are k disjoint common spanning sets?

This statement is known to be true for strongly base orderable matroids [1]; however, the cycle matroid of a graph is not necessarily base orderable.

## References

1. J. Davies, C. McDiarmid, Disjoint common transversals and exchange structures, Journal of the London Mathematical Society s2-14 (1976), 55-62. DOI link.