  An edge labeling of a connected graph $G = (V, E)$ is said to be local
antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any
pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced
vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident
to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$,
is the minimum number of distinct induced vertex labels over all local
antimagic labelings of $G$. In this paper, we first introduce an algorithmic
approach to construct a family of infinitely many even size non-regular
tripartite graphs with $t\ge 1$ component(s) in which every component, called a
{\it Luv} graph, is of odd order $p\ge 9$ and size $q=n(p+1)$ for $n\ge 2$. We
show that every graph in this family has local antimagic chromatic number 3. We
then allowed the $m$-th component to have order $p_m\ge 9$ and size
$n_m(p_m+1)$ for $n_m\ge 2, 1\le m\le t$. We also proved that every such graph
with all components having same order and size also has local antimagic
chromatic number 3. Lastly, we constructed another family of infinitely many
graphs such that different components may have different order and size all of
which having local antimagic chromatic number 3. Consequently, many other
families of (possibly disconnected) graphs with local antimagic chromatic
number 3 are also constructed.
