Julien Bensmail ; Mohammed Senhaji ; Kasper Szabo Lyngsie - On a combination of the 1-2-3 Conjecture and the Antimagic Labelling Conjecture

dmtcs:2168 - Discrete Mathematics & Theoretical Computer Science, August 8, 2017, Vol. 19 no. 1 - https://doi.org/10.23638/DMTCS-19-1-21
On a combination of the 1-2-3 Conjecture and the Antimagic Labelling Conjecture

Authors: Julien Bensmail ; Mohammed Senhaji ; Kasper Szabo Lyngsie

    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 Semaničová-Feňovčíková. 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.


    Volume: Vol. 19 no. 1
    Section: Graph Theory
    Published on: August 8, 2017
    Accepted on: August 8, 2017
    Submitted on: August 8, 2017
    Keywords: 1-2-3 Conjecture,Antimagic Labelling Conjecture,equitable edge-weightings,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
    Fundings :
      Source : OpenAIRE Research Graph
    • Graph Theory: Colourings, flows, and decompositions.; Funder: European Commission; Code: 320812; Call ID: ERC-2012-ADG_20120216; Projet Financing: EC:FP7:ERC

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