On the antimagicness of generalized edge corona graphs

Abstract

Given a graph G, a function of assigning distinct labels $\{1,2,...,|E\left(G\right)\left|\right\}$ to $E\left(G\right)$ such that $w\left(a\right)\ne w\left(b\right)$, ∀ $a,b\in V\left(G\right)$ is an antimagic labeling of G where $w\left(a\right)$ indicates the vertex sum obtained by summing up all the labels assigned to the edges incident on the vertex a. Let G, $H_i$, $1\le i\le m$ be connected graphs such that $E\left(G\right)=\{e_1,e_2,...,e_m\}$. A new graph is constructed from G, $H_i$, $1\le i\le m$ by adding all possible edges between the end vertices of $e_i$ and $V\left(H_i\right)$, $i\in \{1,2,...,m\}$. The resulting graph is called the generalized edge corona of G and $(H_1,H_2,...,H_m)$ which is denoted as $G\diamond (H_1,H_2,...,H_m)$. We prove G ⋄ $(H_1,H_2,...,H_m)$ is antimagic under certain conditions using an algorithmic approach where G has only one vertex of maximum degree three (excluding spider graphs containing uneven legs) and $\left|V\right(H_i\left)\right|\ge 2$, $i\in \{1,2,...,m\}$.

Keywords: Antimagic labeling; Generalized edge corona graphs; Graph labeling; Pan graphs; Spider graphs.