ON LOCAL ANTIMAGIC CHROMATIC NUMBER OF GRAPHS

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 <br /> $omega _{f}(u) neq omega _{f}(v)$ holds; where $omega _{f}(u)=sum _{xin N(u)} f(xu)$. 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 $G$.<br /> 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, <em>Graphs and Combinatorics</em> <strong>33</strong>} (2017), 275-285].