Skip to Main content Skip to Navigation

Further Evidence Towards the Multiplicative 1-2-3 Conjecture

Julien Bensmail 1 Hervé Hocquard 2 Dimitri Lajou 2 Eric Sopena 2
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kaziów 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 Vučković 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.
Document type :
Complete list of metadata

Cited literature [12 references]  Display  Hide  Download
Contributor : Julien Bensmail <>
Submitted on : Friday, April 17, 2020 - 8:40:03 PM
Last modification on : Monday, October 12, 2020 - 10:30:40 AM


Files produced by the author(s)


  • HAL Id : hal-02546401, version 1
  • ARXIV : 2004.09090


Julien Bensmail, Hervé Hocquard, Dimitri Lajou, Eric Sopena. Further Evidence Towards the Multiplicative 1-2-3 Conjecture. [Research Report] Université côte d'azur; Université de bordeaux. 2020. ⟨hal-02546401⟩



Record views


Files downloads