John Haslegrave - Proof of a local antimagic conjecture

dmtcs:3887 - Discrete Mathematics & Theoretical Computer Science, June 4, 2018, Vol. 20 no. 1 -
Proof of a local antimagic conjectureArticle

Authors: John Haslegrave

    An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ such that the sums $S_v=\sum_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \& Semani\v{c}ová-Fe\v{n}ov\v{c}\'iková (2017), and Bensmail, Senhaji \& Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than $K_2$ .

    Volume: Vol. 20 no. 1
    Section: Graph Theory
    Published on: June 4, 2018
    Accepted on: May 14, 2018
    Submitted on: August 29, 2017
    Keywords: Mathematics - Combinatorics,05C78, 05C15
      Source : OpenAIRE Graph
    • Random Graph Geometry and Convergence; Funder: European Commission; Code: 639046

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