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 $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},\mathrm{\mid }E(G)\mathrm{\mid }\}$ 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 $\mathrm{\mid }E(G)\mathrm{\mid }=m$. A generalized edge corona of the graph $G$ and $(H_1,H_2,\mathrm{.}\mathrm{.}\mathrm{.},H_m)$ (denoted by $G\diamond (H_1,H_2,\mathrm{.}\mathrm{.}\mathrm{.}{H}_m)$) is a graph obtained by taking a copy of $G,H_1,H_2,\mathrm{.}\mathrm{.}\mathrm{.},H_m$ and joining the end vertices of $i^{th}$ edge of $G$ to every vertex of $H_i$, $i\in \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},m\}$. 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 $H_i$, $1\le i\le m$ as a connected graph on at least two vertices. We provide an algorithmic approach to prove that $G$ $\diamond$ $(H_1,H_2,\mathrm{.}\mathrm{.}\mathrm{.}{H}_m)$ is antimagic under certain conditions.