An Algorithmic Approach to Antimagic Labeling of Edge Corona Graphs

TL;DR

This paper introduces an algorithmic method to establish antimagic labelings for various complex graphs, including barbell graphs, edge coronas of bistars and regular graphs, and cycles, expanding the class of graphs known to admit such labelings.

Contribution

The paper provides the first algorithmic approach to antimagic labelings for edge corona graphs, demonstrating their applicability to several new graph classes.

Findings

Proves that n-barbell graphs admit antimagic labelings.

Shows edge corona of bistars and regular graphs are antimagic.

Establishes antimagic labelings for cycle coronas.

Abstract

An antimagic labeling of a graph $G$ is a $1-1$ correspondence between the edge set $E(G)$ and $\lbrace 1,2,...,|E(G)|\rbrace$ in which the sum of the labels of edges incident to the distinct vertices are different. The edge corona of any two graphs $G$ and $H$, (denoted by $G$ $\diamond$ $H$) is obtained by joining one copy of $G$ with $|E(G)|$ copies of H such that the end vertices of $i^{th}$ edge of $G$ is adjacent to every vertex in the $i^{th}$ copy of $H$. In this paper, we provide an algorithm to prove that the following graphs admit an antimagic labeling: $-$ $n$-barbell graph $B_n$, $n\geq3$ $-$ edge corona of a bistar graph $B_{x,n}$ and a $k$-regular graph $H$ denoted by $B_{x,n}\diamond H$, $x,n\geq 2$ $-$ edge corona of a cycle $C_m$ and $C_n$ denoted by $C_m \diamond C_n$, $m,n\geq3$