Skip to Main content Skip to Navigation
Reports

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
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
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.
Complete list of metadatas

Cited literature [12 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02546401
Contributor : Julien Bensmail <>
Submitted on : Friday, April 17, 2020 - 8:40:03 PM
Last modification on : Tuesday, May 26, 2020 - 6:50:53 PM

Files

product123.pdf
Files produced by the author(s)

Identifiers

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

Citation

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⟩

Share

Metrics

Record views

50

Files downloads

23