On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture

Julien Bensmail, Mohammed Senhaji, Kasper Szabo Lyngsie

Research output: Contribution to journalJournal articlepeer-review

190 Downloads (Pure)


This paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničov -Feňovč kov . Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices.

Original languageEnglish
Article number21
JournalDiscrete Mathematics and Theoretical Computer Science
Issue number1
Number of pages18
Publication statusPublished - 2017


  • 1-2-3 Conjecture
  • Antimagic Labelling Conjecture
  • Equitable edge-weightings


Dive into the research topics of 'On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture'. Together they form a unique fingerprint.

Cite this