Further evidence towards the multiplicative 1-2-3 Conjecture

Bensmail, Julien; Hocquard, Hervé; Lajou, Dimitri; Sopena, Éric

doi:10.1016/j.dam.2021.10.014

Cited by 5 publications

(2 citation statements)

References 8 publications

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“…In parallel with these results, a product version of the 1-2-3 Conjecture, stating that χ P (G) ≤ 3 for every nice graph G, was introduced by Skowronek-Kaziów in [20]. An interesting fact is that, due to 2 and 3 being coprime, m-proper 3-labellings and p-proper 3-labellings are actually not so distant objects (see [5] for more details), which sort of gave the impression that the product version of the 1-2-3 Conjecture might be of intermediate difficulty, between the sum and multiset versions, yet closer to the latter one. Again, this was confirmed recently in [6], by Bensmail, Hocquard, Lajou and Sopena proving the product version of the 1-2-3 Conjecture in full.…”

Section: Introductionmentioning

confidence: 99%

On proper 2-labellings distinguishing by sums, multisets or products

Bensmail¹,

Fioravantes²

2024

Discuss. Math. Graph Theory

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Given a graph G, a k-labelling ℓ of G is an assignment ℓ : E(G) → {1, . . . , k} of labels from {1, . . . , k} to the edges. We say that ℓ is s-proper, m-proper or p-proper, if no two adjacent vertices of G are incident to the same sum, multiset or product, respectively, of labels.Proper labellings are part of the field of distinguishing labellings, and have been receiving quite some attention over the last decades, in particular in the context of the well-known 1-2-3 Conjecture. In recent years, quite some progress was made towards the main questions of the field, with, notably, the analogues of the 1-2-3 Conjecture for m-proper and p-proper labellings being solved. This followed mainly from a better global understanding of these types of labellings.In this note, we focus on a question raised by Paramaguru and Sampathkumar, who asked whether graphs with m-proper 2-labellings always admit s-proper 2-labellings. A negative answer to this question was recently given by Luiz, who provided infinite families of counterexamples. We give a more general result, showing that recognising graphs with m-proper 2-labellings but no s-proper 2-labellings is an NP-hard problem. We also prove a similar result for m-proper 2-labellings and p-proper 2-labellings, and raise a few directions for further work on the topic.

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Section: Introductionmentioning

confidence: 99%

On proper 2-labellings distinguishing by sums, multisets or products

Bensmail¹,

Fioravantes²

2024

Discuss. Math. Graph Theory

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“…The Multiplicative 1-2-3 Conjecture has received, to date, much less attention than the original 1-2-3 Conjecture. Towards it, the main results we know of are that the conjecture holds for 4-colourable graphs [4], and that χ Π (G) ≤ 4 holds for every nice graph G [15].…”

Section: Introductionmentioning

confidence: 99%

On a List Variant of the Multiplicative 1-2-3 Conjecture

Bensmail¹,

Hocquard²,

Lajou³

et al. 2021

Preprint

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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.

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