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Pré-Publication, Document De Travail Année : 2015

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.
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Dates et versions

hal-01175756 , version 1 (12-07-2015)
hal-01175756 , version 2 (01-10-2015)
hal-01175756 , version 3 (12-09-2016)

Identifiants

  • HAL Id : hal-01175756 , version 1

Citer

Emma Barme, Julien Bensmail. Directed variations of the 1-2-3 Conjecture. 2015. ⟨hal-01175756v1⟩
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