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  1. Home > Articles & Issues >
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  3. Proof of a local ant ...
Article

Proof of a local antimagic conjecture

John Haslegrave ORCID
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Open on arXiv
Submitted on
August 29, 2017
Accepted on
May 14, 2018
Published on
June 4, 2018
Last modified on
March 31, 2025
Volume 20
Vol. 20:1
Graph Theory
DOI
10.23638/DMTCS-20-1-18
License
arXiv.org - Non-exclusive license to distribute
European Commission
639046
Random Graph Geometry and Convergence
Indicators
1648
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1184
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Proof of a local antimagic conjecture

John Haslegrave ORCID
Abstract
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}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (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 $K_2$ . Comment: Final version for publication in DMTCS. Changes from previous version are formatting to journal style and correction of two minor typographical errors
Keywords
  • Mathematics - Combinatorics
  • 05C78, 05C15
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