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

Abstract

An edge labeling of a connected graph $G=(V,E)$ is said to be local antimagic if it is a bijection $f:E\to \{1,\dots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^{+}(x)\ne f^{+}(y)$, 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.