Antimagic Labeling of Generalized Edge Corona Graphs

An antimagic labeling of a graph $G$ is a one-to-one correspondence between the edge set $E(G)$ and $\lbrace 1,2,...,|E(G)|\rbrace$ in which the sum of the edge labels incident on the distinct vertices are distinct. Let $G$,$H_1$,$H_2$,...,$H_{m-1}$, and $H_m$ be simple graphs where $|E(G)|=m$. A generalized edge corona of the graph $G$ and $(H_1,H_2,...,H_m)$ (denoted by $G\diamond (H_1,H_2,...H_m)$) is a graph obtained by taking a copy of $G,H_1,H_2,...,H_m$ and joining the end vertices of $i^{th}$ edge of $G$ to every vertex of $H_i$, $i\in\lbrace 1,2,...,m\rbrace$. In this paper, we consider $G$ as a connected graph with exactly one vertex of maximum degree 3 (excluding the spider graph with exactly one vertex of maximum degree 3 containing uneven legs) and each $H_i$, $1\leq i \leq m$ as a connected graph on at least two vertices. We provide an algorithmic approach to prove that $G$ $\diamond$ $(H_1,H_2,...H_m)$ is antimagic under certain conditions.