# Decomposition of oriented k-partition-connected digraphs

Let *D=(V,A)* be a digraph whose underlying graph is *k*-partition-connected, and let [math]r_0 \in V[/math] 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 [math]v \in V-r_0[/math] has positive in-degree in each subgraph?

## Remarks

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

*Is it true that if both [math]M_1[/math] and [math]M_2[/math] 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.