Proof of Berge's strong path partition conjecture for k = 2

Berge's strong path partition conjecture from 1982 generalizes and extends Dilworth's theorem and the Greene-Kleitman theorem which are well known for partially ordered sets. The conjecture is known to be true for all digraphs only for k = 1 (by the Gallai-Milgram theorem) and for k ≥ λ (where λ is the cardinality of the longest path in the graph). The attempts made, so far, to prove the conjecture for other values of k have yielded proofs for acyclic digraphs, but not for general digraphs. In this paper, we prove the conjecture for k = 2 for all digraphs. The proof is constructive and it extends the proof for k = 1.