An Orientation Theorem with Parity Conditions

Given a graph G = (V, E) and a set T ⊆ V , an orientation of G is called T -odd if precisely the vertices of T get odd in-degree. We give a good characterization for the existence of a T -odd orientation for which there exist k edge-disjoint spanning arborescences rooted at a prespecified set of k roots. Our result implies Nash-Williams’ theorem on covering the edges of a graph by k forests and a (generalization of a) theorem due to Nebeský on upper embeddable graphs.