Skip to Main content Skip to Navigation
Reports

On a List Variant of 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 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with 1,2,3 so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in literature, including more general versions and slight variations. Notable such variations include the List 1-2-3 Conjecture variant, in which edges must be assigned labels from dedicated lists of three labels, and the Multiplicative 1-2-3 Conjecture variant, in which labels 1,2,3 must be assigned to the edges so that adjacent vertices are incident to different products of labels. Several results obtained towards these two variants led to observe some behaviours that are distant from those of the original conjecture. In this work, we consider the list version of the Multiplicative 1-2-3 Conjecture, proposing the first study dedicated to this very problem. In particular, given any graph G, we wonder about the minimum k such that G can be labelled as desired when its edges must be assigned labels from dedicated lists of size k. Exploiting a relationship between our problem and the List 1-2-3 Conjecture, we provide upper bounds on k when G belongs to particular classes of graphs. We further improve some of these bounds through dedicated arguments.
Document type :
Reports
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-03120800
Contributor : Julien Bensmail <>
Submitted on : Monday, January 25, 2021 - 8:45:57 PM
Last modification on : Wednesday, January 27, 2021 - 3:33:41 AM

File

main.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-03120800, version 1

Citation

Julien Bensmail, Hervé Hocquard, Dimitri Lajou, Eric Sopena. On a List Variant of the Multiplicative 1-2-3 Conjecture. [Research Report] Université de bordeaux; Université côte d'azur. 2021. ⟨hal-03120800⟩

Share

Metrics

Record views

32

Files downloads

18