  A {\it local antimagic labeling} of a connected graph $G$ with at least three
vertices, is a bijection $f:E(G) \rightarrow \{1,2,\ldots , |E(G)|\}$ such that
for any two adjacent vertices $u$ and $v$ of $G$, the condition $\omega _{f}(u)
\neq \omega _{f}(v)$ holds; where $\omega _{f}(u)=\sum _{x\in N(u)} f(xu)$.
Assigning $\omega _{f}(u)$ to $u$ for each vertex $u$ in $V(G)$, induces
naturally a proper vertex coloring of $G$; and $|f|$ denotes the number of
colors appearing in this proper vertex coloring. The {\it local antimagic
chromatic number} of $G$, denoted by $\chi _{la}(G)$, is defined as the minimum
of $|f|$, where $f$ ranges over all local antimagic labelings of $G$. In this
paper, we explicitely construct an infinite class of connected graphs $G$ such
that $\chi _{la}(G)$ can be arbitrarily large while $\chi _{la}(G \vee
\bar{K_{2}})=3$, where $G \vee \bar{K_{2}}$ is the join graph of $G$ and the
complement graph of $K_{2}$. This fact leads to a counterexample to a theorem
of [Local antimagic vertex coloring of a graph, {\em Graphs and Combinatorics}\
{\bf 33} (2017), 275--285].

Combinatorics