Proof of a local antimagic conjecture

TL;DR

This paper proves that every connected graph other than K2 admits a local antimagic labelling, confirming a conjecture and establishing the well-definedness of the local antimagic chromatic number for such graphs.

Contribution

It provides a proof of the conjecture that all connected graphs except K2 have a local antimagic labelling, using probabilistic methods.

Findings

Confirmed the conjecture for all connected graphs except K2.

Established the local antimagic chromatic number is well-defined for these graphs.

Used probabilistic techniques to prove the existence of local antimagic labellings.

Abstract

An antimagic labelling of a graph $G$ is a bijection $f:E(G)\to\{1,\ldots,E(G)\}$ 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 $K_2$ .