On the role of 3s for the 1-2-3 Conjecture

Bensmail, Julien; Fioravantes, Foivos; Inerney, Fionn Mc

doi:10.1016/j.tcs.2021.09.023

Cited by 3 publications

(14 citation statements)

References 19 publications

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“…and, hence, ω * (v) = − 4 3 for every 1-vertex v, while ω * (v) = 2 3 for every 2-vertex v. Below, a 3-vertex is said weak if it is adjacent to exactly one 4-vertex and no 5 + -vertex.…”

Section: • Configuration C7mentioning

confidence: 99%

“…This leads to several side questions of interest, some of which have already been investigated in the literature. For instance, let us mention [4], in which the authors investigate the existence of graphs needing lots of 3's in their proper 3-labellings, and [5], in which the authors strive to design proper 3-labellings minimising the sum of assigned labels. Regarding these investigations in [4,5], note that Conjecture 2.1, if true, could provide new results, such as new bounds on some parameters.…”

Section: The Importance Of Label 3 For the 1-2-3 Conjecturementioning

confidence: 99%

“…A related interesting question we have regarding those concerns is about the existence of graphs such that, in all their proper 3-labellings, there must be at least two adjacent edges being assigned label 3. As far as we are aware, no such graphs are known to date, although graphs requiring "lots" of 3's do exist (see [4]). The existence of such graphs would imply that, in some circumstances, labels 1 and 2 only are far from sufficient, as there are local "hard" places where multiple 3's must be placed to get a proper 3-labelling.…”

Section: The Importance Of Label 3 For the 1-2-3 Conjecturementioning

confidence: 99%

“…4 , so that σ(u) = 8 and, eventually, no x-cycle can contain u. Now assign labels in {1, 2} to v 1 v 2 and v 3 v 4 , chosen so that, assuming v ′ 1 and v ′ 4 denote the neighbour of v 1 and v 4 , respectively, not in {u, v 2 , v 3 }, we get…”

mentioning

confidence: 99%

See 3 more Smart Citations

On Inducing Degenerate Sums Through 2-Labellings

Bensmail,

Hocquard,

Marcille

2024

Graphs and Combinatorics

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We deal with a variant of the 1-2-3 Conjecture introduced by Gao, Wang, and Wu in 2016. This variant asks whether all graphs can have their edges labelled with 1 and 2 so that when computing the sums of labels incident to the vertices, no monochromatic cycle appears. In the aforementioned seminal work, the authors mainly verified their conjecture for a few classes of graphs, namely graphs with maximum average degree at most 3 and series-parallel graphs, and observed that it also holds for simple classes of graphs (cycles, complete graphs, and complete bipartite graphs).In this work, we provide a deeper study of this conjecture, establishing strong connections with other, more or less distant notions of graph theory. While this conjecture connects quite naturally to other notions and problems surrounding the 1-2-3 Conjecture, it can also be expressed so that it relates to notions such as the vertex-arboricity of graphs. Exploiting such connections, we provide easy proofs that the conjecture holds for bipartite graphs and 2-degenerate graphs, thus generalising some of the results of Gao, Wang, and Wu. We also prove that the conjecture holds for graphs with maximum average degree less than 10 3 , thereby strengthening another of their results. Notably, this also implies the conjecture holds for planar graphs with girth at least 5. All along the way, we also raise observations and results highlighting why the conjecture might be of greater interest.

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Section: • Configuration C7mentioning

confidence: 99%

Section: The Importance Of Label 3 For the 1-2-3 Conjecturementioning

confidence: 99%

Section: The Importance Of Label 3 For the 1-2-3 Conjecturementioning

confidence: 99%

mentioning

confidence: 99%

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On Inducing Degenerate Sums Through 2-Labellings

Bensmail,

Hocquard,

Marcille

2024

Graphs and Combinatorics

Self Cite

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show abstract

“…We have thus reached a point of time where quite some progress towards the 1-2-3 Conjecture has been recently made, through the upper bound, 5, being very close to what is conjectured exactly, and two related conjectures, which for quite some time seemed of close hardness, being proved. What made this recent progress possible, is definitely a better understanding over the inherent behaviour of s-proper, m-proper and p-proper 3-labellings, which on themselves, gave birth to interesting side investigations (see [3] and the references there for examples). As a matter of fact, even if the 1-2-3 Conjecture turned out to be proven soon, there would still remain lots of interesting questions to answer towards fully understanding proper labellings.…”

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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Maximising 1’s through proper labellings

Bensmail

2024

Discuss. Math. Graph Theory

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We investigate graph proper labellings, i.e., assignments of labels to the edges so that no two adjacent vertices are incident to the same sum of labels, with the additional requirement that label 1 must be assigned to as many edges as possible. The study of such objects is motivated by practical concerns, and by connections with other types of proper labellings in which other additional properties (such as minimising the sum of assigned labels, or minimising the use of label 3) must be met. We prove that maximising 1's is a problem on its own, in that it is not equivalent to any of these other labelling problems with optimisation concerns. We then provide labelling tools and techniques for designing proper labellings with many 1's. As a result, we prove that, for several graph classes, it is always possible to design proper labellings where label 1 is assigned to about half the edges.

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On the role of 3s for the 1-2-3 Conjecture

Cited by 3 publications

References 19 publications

On Inducing Degenerate Sums Through 2-Labellings

On Inducing Degenerate Sums Through 2-Labellings

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

Maximising 1’s through proper labellings

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