On local antimagic chromatic number of cycle-related join graphs II

TL;DR

This paper investigates the local antimagic chromatic number of join graphs, providing conditions and exact values for many such graphs, advancing understanding of graph labelings.

Contribution

It offers new sufficient conditions and exact calculations for the local antimagic chromatic number of join graphs, expanding prior knowledge in graph labeling theory.

Findings

Derived sufficient conditions for the local antimagic chromatic number of join graphs.

Determined exact values of the local antimagic chromatic number for numerous join graphs.

Enhanced understanding of how join operations affect local antimagic labelings.

Abstract

An edge labeling of a 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 of $x$ is $f^+(x)= \sum_{e\in E(x)} f(e)$ ($E(x)$ is the set of 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, several sufficient conditions to determine the local antimagic chromatic number of the join of graphs are obtained. We then determine the exact value of the local antimagic chromatic number of many join graphs.