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 $g : E \to \{1,2,\ldots,q\}$ 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 $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ 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.