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 →\1,… ,|E|\ such that for any pair of adjacent vertices x and y, f⁺(x)= f⁺(y), where the induced vertex label f⁺(x)= ∑ f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χₗₐ(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. Suppose χₗₐ(G)=χₗₐ(H) and GH 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≥ 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.