On join product and local antimagic chromatic number of regular graphs

Abstract

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$, where $f^+(u) = \sum_{e\in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $f^+(v)$. The local antimagic chromatic number, denoted $\chi_{la}(G)$, is the minimum number of induced colors taken over local antimagic labeling of $G$. Let $G$ and $H$ be two vertex disjoint graphs. The join graph of $G$ and $H$, denoted $G \vee H$, is the graph $V(G\vee H) = V(G) \cup V(H)$ and $E(G\vee H) = E(G) \cup E(H) \cup \{uv \,|\, u\in V(G), v \in V(H)\}$. In this paper, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers.