  Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A
graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A
bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of
$G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$,
where $f^+(u) = \sum_{e\in E(u)} f(e)$, and $E(u)$ is the set of edges incident
to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of
$G$ if vertex $v$ is assigned the color $f^+(v)$. The local antimagic chromatic
number, denoted $\chi_{la}(G)$, is the minimum number of induced colors taken
over local antimagic labeling of $G$. Let $G$ and $H$ be two vertex disjoint
graphs. The {\it lexicographic product} of $G$ and $H$, denoted $G[H]$, is the
graph with vertex set $V(G) \times V(H)$, and $(u,u')$ is adjacent to $(v,v')$
in $G[H]$ if $(u,v)\in E(G)$ or if $u=v$ and $u'v'\in E(H)$. In this paper, we
obtained sharp upper bound of $\chi_{la}(G[O_n])$ where $O_n$ is a null graph
of order $n\ge 1$. Sufficient conditions for even regular bipartite and
tripartite graphs $G$ to have $\chi_{la}(G)=3$ are also obtained. Consequently,
we successfully determined the local antimagic chromatic number of infinitely
many (connected and disconnected) regular graphs that partially support the
existence of $r$-regular graph $G$ of order $p$ such that (i)
$\chi_{la}(G)=\chi(G)=k$, and (ii) $\chi_{la}(G)=\chi(G)+1=k$ for each possible
$r,p,k$.
