Super H -Antimagic Total Covering for Generalized Antiprism and Toroidal Octagonal Map

Abstract

Let $G$ be a graph and $H\subseteq G$ be subgraph of $G$ . The graph $G$ is said to be $\left(a,d\right)$ - $H$ antimagic total graph if there exists a bijective function $f:V\left(H\right)\cup E\left(H\right)⟶\left\{\mathrm{1,2,3},\dots ,\left|V\left(H\right)\right|+\left|E\left(H\right)\right|\right\}$ such that, for all subgraphs isomorphic to $H$ , the total $H$ weights $W\left(H\right)=W\left(H\right)={\displaystyle {\sum }_{x\in V\left(H\right)}f\left(x\right)}+{\displaystyle {\sum }_{y\in E\left(H\right)}f\left(y\right)}$ forms an arithmetic sequence $a,a+d,a+2d,\dots ,a+\left(n-1\right)d$ , where $a$ and $d$ are positive integers and $n$ is the number of subgraphs isomorphic to $H$ . An $\left(a,d\right)$ - $H$ antimagic total labeling $f$ is said to be super if the vertex labels are from the set $\left\{\mathrm{1,2},\dots ,|V\left(G\right)\right\}$ . In this paper, we discuss super $\left(a,d\right)$ - $C_3$ -antimagic total labeling for generalized antiprism and a super $\left(a,d\right)$ - $C_8$ -antimagic total labeling for toroidal octagonal map.