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

Julien Bensmail 1 Foivos Fioravantes 1
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : 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. We give a negative answer to this question, 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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https://hal.archives-ouvertes.fr/hal-03536425
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Submitted on : Wednesday, January 19, 2022 - 10:18:08 PM
Last modification on : Wednesday, February 2, 2022 - 11:42:55 AM
Long-term archiving on: : Wednesday, April 20, 2022 - 7:28:05 PM

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Julien Bensmail, Foivos Fioravantes. On proper 2-labellings distinguishing by sums, multisets or products. [Research Report] Université Côte d'Azur. 2022. ⟨hal-03536425⟩

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