Olivier Baudon ; Julien Bensmail ; Jakub Przybyło ; Mariusz Woźniak - On locally irregular decompositions and the 1-2 Conjecture in digraphs

dmtcs:4474 - Discrete Mathematics & Theoretical Computer Science, October 1, 2018, vol. 20 no. 2 - https://doi.org/10.23638/DMTCS-20-2-7
On locally irregular decompositions and the 1-2 Conjecture in digraphs

Authors: Olivier Baudon ; Julien Bensmail ; Jakub Przybyło ORCID-iD; Mariusz Woźniak

    The 1-2 Conjecture raised by Przybylo and Wozniak in 2010 asserts that every undirected graph admits a 2-total-weighting such that the sums of weights "incident" to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally irregular decompositions) to digraphs, we here introduce and study several variants of the 1-2 Conjecture for digraphs. For every such variant, we raise conjectures concerning the number of weights necessary to obtain a desired total-weighting in any digraph. We verify some of these conjectures, while we obtain close results towards the ones that are still open.


    Volume: vol. 20 no. 2
    Section: Graph Theory
    Published on: October 1, 2018
    Accepted on: October 1, 2018
    Submitted on: April 30, 2018
    Keywords: 1-2 Conjecture,locally irregular decompositions,digraphs, [ 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
    • Structures Interdites; Funder: French National Research Agency (ANR); Code: ANR-13-BS02-0007

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