On a directed variation of the 1-2-3 and 1-2 Conjectures

Emma Barme 1 Julien Bensmail 2, 3 Jakub Przybyło 4 Mariusz Woźniak 4
1 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
3 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : In this paper, we consider the following question, which stands as a directed analogue of the well-known 1-2-3 Conjecture: Given any 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? We answer positively to this question, and investigate digraphs for which even the weights among {1,2} are sufficient. In relation with the so-called 1-2 Conjecture, we also consider a total version of the problem, which we prove to be false. Our investigations turn to have interesting relations with open questions related to the 1-2-3 Conjecture.
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Emma Barme, Julien Bensmail, Jakub Przybyło, Mariusz Woźniak. On a directed variation of the 1-2-3 and 1-2 Conjectures. Discrete Applied Mathematics, Elsevier, 2017, 217 (2), pp.123-131. ⟨hal-01175756v3⟩

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