Geodesic
On locally irregular decompositions and the 1-2 Conjecture in digraphs
Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2.

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The 1-2 Conjecture raised by Przybylo and Wozniak in 2010 asserts that every undirected graph admits a 2-total-weighting such that the sums of weights "incident" to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally irregular decompositions) to digraphs, we here introduce and study several variants of the 1-2 Conjecture for digraphs. For every such variant, we raise conjectures concerning the number of weights necessary to obtain a desired total-weighting in any digraph. We verify some of these conjectures, while we obtain close results towards the ones that are still open. 
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     author = {Baudon , Olivier and Bensmail , Julien and Przyby{\l}o , Jakub and Wo\'zniak , Mariusz},
     title = {On locally irregular decompositions and the 1-2 {Conjecture} in digraphs},
     journal = {Discrete mathematics & theoretical computer science},
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Baudon , Olivier; Bensmail , Julien; Przybyło , Jakub; Woźniak , Mariusz. On locally irregular decompositions and the 1-2 Conjecture in digraphs. Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2. doi : 10.23638/DMTCS-20-2-7. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-2-7/

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