On Local Antimagic Chromatic Number of Graphs

TL;DR

This paper constructs an infinite class of graphs with arbitrarily large local antimagic chromatic number, yet their join with a complement of K2 has chromatic number 3, providing a counterexample to a previous theorem.

Contribution

It explicitly constructs graphs demonstrating that local antimagic chromatic number can be arbitrarily large while their join with a specific graph has chromatic number 3, challenging prior assumptions.

Findings

Constructed infinite graphs with arbitrarily large local antimagic chromatic number.

Showed that the join of these graphs with the complement of K2 has chromatic number 3.

Provided a counterexample to a previously published theorem.

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 explicitely 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 $K_{2}$. This fact leads to a counterexample to a theorem of [Local antimagic vertex coloring of a graph, {\em Graphs and Combinatorics}\ {\bf 33} (2017), 275--285].