A Generalized Version of a Local Antimagic Labelling Conjecture

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).