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Journal Articles Discrete Applied Mathematics Year : 2017

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

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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Dates and versions

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

Identifiers

  • HAL Id : hal-01175756 , version 3

Cite

Emma Barme, Julien Bensmail, Jakub Przybyło, Mariusz Woźniak. On a directed variation of the 1-2-3 and 1-2 Conjectures. Discrete Applied Mathematics, 2017, 217 (2), pp.123-131. ⟨hal-01175756v3⟩
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