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.