Every Graph Is Local Antimagic Total

Gee-Choon Lau

Published 2019-06-25Version 1

An antimagic labelling of a graph $G$ is a bijection $h : E(G) \to \{1, \ldots, |E(G)|\}$ such that the induced vertex label $h^+(v) = \sum_{uv\in E(G)}$ distinguish all vertices $v$. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than $K_2$ admits an antimagic labelling. In 2017, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \& Semani\v{c}ov\'{a}-Fe\v{n}ov\v{c}\'{i}kov\'{a} and Bensmail, Senhaji \& Szabo Lyngsie) 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. Recently, Haslegrave (2018) proved that every graph without isolated edges admits a local antimagic labeling. A graph $G = (V, E)$ is said to be local antimagic total if there exists a bijection $f: V\cup E \to\{1,2,\ldots ,|V\cup E|\}$ such that for any pair of adjacent vertices $u$ and $v$, $w(u)\not= w(v)$, where the induced vertex weight $w(u)= f(u) +\sum f(e)$, with $e$ ranging over all the edges incident to $u$. The local antimagic total chromatic number of $G$, denoted by $\chi_{lat}(G)$, is the minimum number of distinct induced vertex weights over all local antimagic total labelings of $G$. In this note, we proved that every graph admits a local antimagic total labeling. We then determine the exact value of $\chi_{lat}(G)$ for some standard graphs $G$.