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

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. 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.

On the algorithmic complexity of determining the AVD and NSD chromatic indices of graphs

“…Our main results in this work deal precisely with such forcing mechanisms. For this reason, it is not surprising that some notions and gadgets introduced through what follows are, in spirit at least, reminiscent of ones designed for related different problems (such as those from [1]).…”

“…Our first intuitions on that question were inspired from [1], in which similar questions were answered for non-proper versions of AVD and NSD edge-colourings. In brief, therein, the authors consider k-labellings, which are, essentially, k-edge-colourings (no properness condition is required whatsoever).…”

“…In [1], the authors answered this question negatively, showing that, for a graph G with χ M (G) = 2, it is NP-hard to decide whether χ S (G) = 2. This shows that there exist infinitely many graphs G with χ M (G) = χ S (G).…”