Construction of local antimagic 3-colorable graphs of fixed even size -- matrix approach

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 $G$. Suppose $\chi_{la}(G)=\chi_{la}(H)$ and $G_H$ is obtained from $G$ and $H$ by merging some vertices of $G$ with some vertices of $H$ bijectively. In this paper, we give ways to construct matrices with integers in $[1,10k]$, $k\ge 1$, that meet certain properties. Consequently, we obtained many families of (disconnected) bipartite (and tripartite) graphs of size $10k$ with local antimagic chromatic number 3.