A general decomposition theory for the 1-2-3 Conjecture and locally irregular decompositions

Olivier Baudon 1 Julien Bensmail 2 Tom Davot 3 Hervé Hocquard 1 Jakub Przybyło 4 Mohammed Senhaji 1 Eric Sopena 1 Mariusz Woźniak 4
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
3 MAORE - Méthodes Algorithmes pour l'Ordonnancement et les Réseaux
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : How can one distinguish the adjacent vertices of a graph through an edge-weighting? In the last decades, this question has been attracting increasing attention, which resulted in the active field of distinguishing labellings. One of its most popular problems is the one where neighbours must be distinguishable via their incident sums of weights. An edge-weighting verifying this is said neighbour-sum-distinguishing. The popularity of this notion arises from two reasons. A first one is that designing a neighbour-sum-distinguishing edge-weighting showed up to be equivalent to turning a simple graph into a locally irregular (i.e., without neighbours with the same degree) multigraph by adding parallel edges, which is motivated by the concept of irregularity in graphs. Another source of popularity is probably the influence of the famous 1-2-3 Conjecture, which claims that such weightings with weights in {1,2,3} exist for graphs with no isolated edge. The 1-2-3 Conjecture has recently been investigated from a decompositional angle, via so-called locally irregular decompositions, which are edge-partitions into locally irregular subgraphs. Through several recent studies, it was shown that this concept is quite related to the 1-2-3 Conjecture. However, the full connexion between all those concepts was not clear. In this work, we propose an approach that generalizes all concepts above, involving coloured weights and sums. As a consequence, we get another interpretation of several existing results related to the 1-2-3 Conjecture. We also come up with new related conjectures, to which we give some support.
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[Research Report] LaBRI, Université de Bordeaux; I3S, Université Côte d'Azur; AGH University of Science and Technology; LIRMM, Université de Montpellier. 2018, pp.1-16
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Olivier Baudon, Julien Bensmail, Tom Davot, Hervé Hocquard, Jakub Przybyło, et al.. A general decomposition theory for the 1-2-3 Conjecture and locally irregular decompositions. [Research Report] LaBRI, Université de Bordeaux; I3S, Université Côte d'Azur; AGH University of Science and Technology; LIRMM, Université de Montpellier. 2018, pp.1-16. 〈hal-01690222v2〉

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