On local antimagic chromatic number of cycle-related join graphs

TL;DR

This paper investigates the local antimagic chromatic number of cycle-related join graphs, providing conditions for its bounds and determining exact values for many such graphs.

Contribution

It offers new sufficient conditions for comparing local antimagic chromatic numbers and calculates exact values for a class of cycle-related join graphs.

Findings

Established bounds for $ ext{chi}_{la}(H)$ relative to $ ext{chi}_{la}(G)$.

Derived exact local antimagic chromatic numbers for various cycle-related join graphs.

Provided conditions under which the chromatic number remains invariant or bounded when edges are added or removed.

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, several sufficient conditions for $\chi_{la}(H)\le \chi_{la}(G)$ are obtained, where $H$ is obtained from $G$ with a certain edge deleted or added. We then determined the exact value of the local antimagic chromatic number of many cycle related join graphs.