Affirmative Solutions On Local Antimagic Chromatic Number

TL;DR

This paper investigates the local antimagic chromatic number of graphs, providing counterexamples, establishing sharp bounds, and completely determining this number for complete bipartite graphs, advancing understanding of graph labelings.

Contribution

It offers new bounds and conditions for the local antimagic chromatic number, corrects previous lower bounds, and fully characterizes this number for complete bipartite graphs.

Findings

Counterexamples to previous lower bounds.

Sharp lower bounds for $oldsymbol{oldsymbol{ ext{chi}}_{la}(G owtie O_n)}$.

Complete determination of $oldsymbol{oldsymbol{ ext{chi}}_{la}}$ for complete bipartite graphs.

Abstract

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we give counterexamples to the lower bound of $\chi_{la}(G \vee O_2)$ that was obtained in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33 : 275 - 285 (2017)]. A sharp lower bound of $\chi_{la}(G\vee O_n)$ and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs.