Proof of a local antimagic conjecture

An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to \{1,\dots ,E(G)\}$ such that the sums $S_v={\sum }_{e\ni v}f(e)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a & Semani\v{c}ov'a-Fe\v{n}ov\v{c}'ikov'a (2017), and Bensmail, Senhaji & Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than $K_2$ admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than $K_2$ .