On local antimagic total labeling of complete graphs amalgamation

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 $ 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. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of $K_1$ and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.