Local antimagic chromatic number of partite graphs

Abstract

Let $G$ be a connected graph with $|V|=n$ and $|E|=m$. A bijection $f:E\to \{1,2,...,m\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, $w(u)\ne w(v)$, where $w(u)={\sum }_{e\in E(u)}f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $v$ is assigned the color $w(v)$. The local antimagic chromatic number is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. Let $m,n>1$. In this paper, the local antimagic chromatic number of a complete tripartite graph $K_{1,m,n}$, and $r$ copies of a complete bipartite graph $K_{m,n}$ where $m\not\equiv n\mathrm{mod}2$ are determined.