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Degree sum conditions for oriented forests in digraphs. (English)
Discrete Math. 309, No. 13, 4642-4645 (2009).
Summary: 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$\{d^{+}(u)+d^{+}(v),d^{ - }(u)+d^{ - }(v),d^{+}(u)+d^{ - }(v)\}\geq 2m - 1$ for every pair of vertices $u,v\in V(D)$ with $uv\notin A(D)$. This is a common generalization of two results of {\it Ch.\,S. Babu} and {\it A.\,A. Diwan} [“Degree conditions for forests in graphs,” Discrete Math. 301, No.\,2-3, 228‒231 (2005; Zbl 1078.05020); “Oriented forests in directed graphs,” Electron. Notes Discrete Math. 22, 141-145 (2005; Zbl 1182.05057)], 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.