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

dmtcs:4682 - Discrete Mathematics & Theoretical Computer Science, April 2, 2019, vol. 21 no. 1, ICGT 2018 - https://doi.org/10.23638/DMTCS-21-1-2
A general decomposition theory for the 1-2-3 Conjecture and locally irregular decompositionsArticle

Authors: Olivier Baudon 1; Julien Bensmail 2; Tom Davot ORCID3; Hervé Hocquard ORCID1; Jakub Przybyło ORCID4; Mohammed Senhaji 1; Eric Sopena ORCID1; Mariusz Woźniak 4

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.

Volume: vol. 21 no. 1, ICGT 2018
Published on: April 2, 2019
Accepted on: March 25, 2019
Submitted on: July 13, 2018
Keywords: Coloured weighted degrees,1-2-3 Conjecture,Locally irregular decompositions,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]

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