On locally irregular decompositions and the 1-2 Conjecture in digraphs

Abstract : 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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Olivier Baudon, Julien Bensmail, Jakub Przybyło, Mariusz Woźniak. On locally irregular decompositions and the 1-2 Conjecture in digraphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, In press. ⟨hal-01374427v4⟩

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