On locally irregular decompositions and the 1-2 Conjecture in digraphs - Archive ouverte HAL
Journal Articles Discrete Mathematics and Theoretical Computer Science Year : 2018

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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Dates and versions

hal-01374427 , version 1 (30-09-2016)
hal-01374427 , version 2 (30-04-2018)
hal-01374427 , version 3 (23-08-2018)
hal-01374427 , version 4 (11-09-2018)
hal-01374427 , version 5 (29-09-2018)

Identifiers

Cite

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, 2018, vol. 20 no. 2, ⟨10.23638/DMTCS-20-2-7⟩. ⟨hal-01374427v5⟩
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