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It is known that null graphs and 1-regular graphs are the only regular graphs without local antimagic chromatic number. In this paper, we proved that the join of 1-regular graph and a null graph has local antimagic chromatic number is 3. Consequently, we also obtained many families of (possibly disconnected or regular) bipartite and tripartite graph with local antimagic chromatic number 3.

For a graph $G(V,E)$ of size $q$, a bijection $f : E(G) \to [1,q]$ is a local antimagc labeling if it induces a vertex labeling $f^+ : V(G) \to \mathbb{N}$ such that $f^+(u) \ne f^+(v)$, where $f^+(u)$ is the sum of all the incident edge label(s) of $u$, for every edge $uv \in E(G)$. In this paper, we make use of matrices of fixed sizes to construct several families of infinitely many tripartite graphs with local antimagic chromatic number 3., Comment: arXiv admin note: text overlap with arXiv:2408.04942

It is known that null graphs and 1-regular graphs are the only regular graphs without local antimagic chromatic number. In this paper, we use matrices of size $(2m+1) \times (2k+1)$ to completely determine the local antimagic chromatic number of the join of null graphs, $O_m, m\ge 1,$ and 1-regular graphs of odd components, $(2k+1)P_2$, $k\ge 1$. Consequently, we obtained infinitely many (possibly disconnected or regular) tripartite graphs with local antimagic chromatic number 3., Comment: 15 pages, 8 figures

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.

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if there is a bijection $f:E \to\{1,\ldots ,|E|\}$ 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 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.