Degree sum conditions for oriented forests in digraphs

Let F be an oriented forest with n vertices and m arcs and D be a digraph without loops and multiple arcs. In this note we prove that D contains a subdigraph isomorphic to F if D has at least n vertices and min {d⁺ (u) + d⁺ (v), d^- (u) + d^- (v), d⁺ (u) + d^- (v)} ≥ 2 m - 1 for every pair of vertices u, v ∈ V (D) with u v ∉ A (D). This is a common generalization of two results of Babu and Diwan, one on the existence of forests in graphs under a degree sum condition and the other on the existence of oriented forests in digraphs under a minimum degree condition.