A Generalized Version of a Local Antimagic Labelling Conjecture

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 u, v 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 u, v 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 u, v we have that $$\sum _{e \in E(v)}f(e) \ne \sum _{e \in E(u)}f(e)$$ .