The direct product of a star and a path is antimagic

TL;DR

This paper investigates the antimagic labeling property for a specific class of disconnected graphs formed by the direct product of a star and a path, contributing to the understanding of antimagicness in complex graph structures.

Contribution

It proves that the direct product of a star and a path graph admits an antimagic labeling, advancing knowledge on antimagic properties of disconnected graphs.

Findings

The direct product of a star and a path is antimagic.

Provides a constructive method for antimagic labeling in this class.

Extends antimagic graph theory to new disconnected graph classes.

Abstract

A graph $G$ is antimagic if there exists a bijection $f$ from $E(G)$ to $\left\{1,2, \dots,|E(G)|\right\}$ such that the vertex sums for all vertices of $G$ are distinct, where the vertex sum is defined as the sum of the labels of all incident edges. Hartsfield and Ringel conjectured that every connected graph other than $K_2$ admits an antimagic labeling. It is still a challenging problem to address antimagicness in the case of disconnected graphs. In this paper, we study antimagicness for the disconnected graph that is constructed as the direct product of a star and a path.