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arXiv · 2004.09090

Further Evidence Towards the Multiplicative 1-2-3 Conjecture

Julien Bensmail·Hervé Hocquard·Dimitri Lajou·Eric Sopena
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Abstract

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{\'o}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support to the conjecture, it was also proved that all graphs admit such 4-labellings. In this work, we investigate how a recent proof of the multiset version of the 1-2-3 Conjecture by Vu{\v c}kovi{\'c} can be adapted to prove results on the product version. We prove that 4-chromatic graphs verify the product version of the 1-2-3 Conjecture. We also prove that for all graphs we can design 3-labellings that almost have the desired property. This leads to a new problem, that we solve for some graph classes.

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Julien Bensmail, Hervé Hocquard, Dimitri Lajou, Eric Sopena. 2020-04-20. Further Evidence Towards the Multiplicative 1-2-3 Conjecture. https://arxiv.org/abs/2004.09090

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