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A Generalized Version of a Local Antimagic Labelling Conjecture

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Abstract

An antimagic labelling of a graph G with m edges is a bijection \(f: E(G) \rightarrow \{1,\ldots ,m\}\) such that for any two distinct vertices uv we have \(\sum _{e \in E(v)}f(e) \ne \sum _{e \in E(u)}f(e)\), 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 \(f: E(G) \rightarrow \{1,\ldots ,|E(G)|\}\) such that for any two neighbours uv we have \(\sum _{e \in E(v)}f(e) \ne \sum _{e \in E(u)}f(e)\). 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 \(f:E(G) \rightarrow L\) such that for any pair of neighbours uv we have that \(\sum _{e \in E(v)}f(e) \ne \sum _{e \in E(u)}f(e)\).

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Correspondence to Liang Zhong.

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Lyngsie, K.S., Zhong, L. A Generalized Version of a Local Antimagic Labelling Conjecture. Graphs and Combinatorics 34, 1363–1369 (2018). https://doi.org/10.1007/s00373-018-1936-2

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  • DOI: https://doi.org/10.1007/s00373-018-1936-2

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