Document Type : Original Manuscript

Author

School of Mathematics and Computer Science, Damghan University, P.O. Box 36716-41167, Damghan, Iran.

Abstract

A {\it local antimagic labeling} of a connected graph $G$ with at least three vertices, is a bijection $f:E(G) \rightarrow \{1,2,\ldots , |E(G)|\}$ such that for any two adjacent vertices $u$ and $v$ of $G$, the condition
$\omega _{f}(u) \neq \omega _{f}(v)$ holds; where $\omega _{f}(u)=\sum _{x\in N(u)} f(xu)$. Assigning $\omega _{f}(u)$ to $u$ for each vertex $u$ in $V(G)$, induces naturally a proper vertex coloring of $G$; and $|f|$ denotes the number of colors appearing in this proper vertex coloring. The {\it local antimagic chromatic number} of $G$, denoted by $\chi _{la}(G)$, is defined as the minimum of $|f|$, where $f$ ranges over all local antimagic labelings of $G$.
In this paper, we explicitly construct an infinite class  of connected graphs $G$ such that $\chi _{la}(G)$ can be arbitrarily large while $\chi _{la}(G \vee \bar{K_{2}})=3$, where $G \vee \bar{K_{2}}$ is the join graph of $G$ and the complement graph of $K_{2}$. The aforementioned fact leads us to an infinite class of counterexamples to a result of [Local antimagic vertex  coloring of a graph,  Graphs and Combinatorics 33} (2017), 275-285].

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