On local antimagic total labeling of amalgamation graphs

Abstract

Let $G=(V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g:E\to \{1,2,\dots ,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $g^{+}(u)\ne g^{+}(v)$, where $g^{+}(u)={\sum }_{e\in E(u)}g(e)$, and $E(u)$ is the set of edges incident to $u$. Similarly, a bijection $f:V(G)\cup E(G)\to \{1,2,\dots ,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w_f(u)\ne w_f(v)$, where $w_f(u)=f(u)+{\sum }_{e\in E(u)}f(e)$. Thus, any local antimagic (total) labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $g^{+}(v)$ (respectively, $w_f(u)$). The local antimagic (total) chromatic number, denoted ${\chi }_{la}(G)$ (respectively ${\chi }_{lat}(G)$), is the minimum number of induced colors taken over local antimagic (total) labeling of $G$. In this paper, we determined ${\chi }_{lat}(G)$ where $G$ is the amalgamation of complete graphs.