Constructions of local antimagic 3-colorable graphs of fixed odd size | matrix approach

An edge labeling of a connected graph $G=(V,E)$ is said to be local antimagic if there is a bijection $f:E\to \{1,\dots ,\mathrm{\mid }E\mathrm{\mid }\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^{+}(x)≠f^{+}(y)$, 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 give three ways to construct a $(3m+2)\times (2k+1)$ matrix that meets certain properties for $m=1,3$ and $k\ge 1$. Consequently, we obtained many (disconnected) graphs of size $(3m+2)(2k+1)$ with local antimagic chromatic number 3.