Directed variations of the 1-2-3 Conjecture
Résumé
In this paper, we consider the following question, which stands as a directed analogue of the well-known 1-2-3 Conjecture: Given a digraph D with no arc (u,v) verifying d+(u)=d−(v)=1, is it possible to weight the arcs of D with weights among {1,2,3} so that, for every arc (u,v) of D, the sum of incident weights outgoing from u is different from the sum of incident weights incoming to v? Towards this question, we first verify it for D belonging to particular classes of digraphs, before then proving its weakening where 3 is replaced by some absolute constant, namely 17. In the same spirit, we investigate a total version of the same question inspired by the 1-2 Conjecture, and prove that even less weights are necessary in this context, namely 10.
Domaines
Mathématique discrète [cs.DM]
Origine : Fichiers produits par l'(les) auteur(s)