On Path Partitions and Colourings in Digraphs

We provide a new proof of a theorem of Saks which is an extension of Greene’s Theorem to acyclic digraphs, by reducing it to a similar, known extension of Greene and Kleitman’s Theorem. This suggests that the Greene-Kleitman Theorem is stronger than Greene’s Theorem on posets. We leave it as an open question whether the same holds for all digraphs, that is, does Berge’s conjecture concerning path partitions in digraphs imply the extension of Greene’s theorem to all digraphs (conjectured by Aharoni, Hartman and Hoffman)?

• Type: Conference
• Year: 2009
• Where: BIRTHDAY
• Authors: Irith Ben-Arroyo Hartman
• Keywords: Acyclic Digraphs, Applied Computing, BIRTHDAY 2009, Greene-Kleitman Theorem, Open Question