A Generalized Version of a Local Antimagic Labelling Conjecture

Kasper Szabo Lyngsie, Liang Zhong*

*Corresponding author for this work

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Abstract

An antimagic labelling of a graph G with m edges is a bijection f: E(G) → { 1 , … , m} such that for any two distinct vertices u, v we have ∑ e E ( v )f(e) ≠ ∑ e E ( u )f(e) , where E(v) denotes the set of edges incident v. The well-known Antimagic Labelling Conjecture formulated in 1994 by Hartsfield and Ringel states that any connected graph different from K2 admits an antimagic labelling. A weaker local version which we call the Local Antimagic Labelling Conjecture says that if G is a graph distinct from K2, then there exists a bijection f: E(G) → { 1 , … , | E(G) | } such that for any two neighbours u, v we have ∑ e E ( v )f(e) ≠ ∑ e E ( u )f(e). This paper proves the following more general list version of the local antimagic labelling conjecture: Let G be a connected graph with m edges which is not a star. For any list L of m distinct real numbers, there is a bijection f: E(G) → L such that for any pair of neighbours u, v we have that ∑ e E ( v )f(e) ≠ ∑ e E ( u )f(e).

Original languageEnglish
JournalGraphs and Combinatorics
Volume34
Issue number6
Pages (from-to)1363-1369
ISSN0911-0119
DOIs
Publication statusPublished - 1 Nov 2018

Keywords

  • Antimagic labelling
  • Local antimagic labelling
  • Neighbour sum distinguishing edge weightings

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