A Proof of the Multiplicative 1-2-3 Conjecture

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

doi:10.1007/978-3-030-95018-7_1

Cited by 1 publication

(1 citation statement)

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“…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: 52%

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: 52%