On Local Antimagic Vertex Coloring for Corona Products of Graphs

Abstract

Let $G = (V, E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f : E \rightarrow \{1, 2,\cdots, |E|\}$ is called a {\bf local antimagic labeling} if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i.e. $w(u) \neq w(v)$, where the vertex sum $w(u) = \sum_{e \in E(u)} f(e)$, 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(vertex sum) $w(v)$. The {\bf local antimagic chromatic number} $\chi_{la}(G)$ is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. In this article among others we determine completely the local antimagic chromatic number $\chi_{la}(G\circ \overline{K_m})$ for the corona product of a graph $G$ with the null graph $\overline{K_m}$ on $m\geq 1$ vertices, when $G$ is a path $P_n$, a cycle $C_n$, and a complete graph $K_n$.