A Proof of a Conjecture of Bondy Concerning Paths in Weighted Digraphs

Our aim in this note is to prove a conjecture of Bondy, extending a classical theorem of Dirac to edge-weighted digraphs: if every vertex has out-weight at least 1 then the digraph contains a path of weight at least 1. We also give several related conjectures and results concerning heavy cycles in edge-weighted digraphs.