## 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 K_{2} 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 K_{2}, 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 language | English |
---|---|

Journal | Graphs and Combinatorics |

Volume | 34 |

Issue number | 6 |

Pages (from-to) | 1363-1369 |

ISSN | 0911-0119 |

DOIs | |

Publication status | Published - 1 Nov 2018 |

## Keywords

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