On H-antimagic coverings for m-shadow and closed m-shadow of connected graphs

An $(a,d)$ -H-antimagic total labeling of a simple graph G admitting an H-covering is a bijection $\phi :V\left(G\right)\cup E\left(G\right)\to \{1,2,\dots ,|V\left(G\right)|+|E\left(G\right)\left|\right\}$ such that for all subgraphs $H^{\text{'}}$ of G isomorphic to H, the set of $H^{\text{'}}$ -weights given by $w{t}_{\phi }\left(H^{\text{'}}\right)={\sum }_{v\in V\left(H^{\text{'}}\right)}\phi \left(v\right)+{\sum }_{e\in E\left(H^{\text{'}}\right)}\phi \left(e\right)$ forms an arithmetic sequence $a,a+d,\dots ,a+(t-1)d$ where $a>0$ , $d⩾0$ are two fixed integers and t is the number of all subgraphs of G isomorphic to H. Moreover, such a labeling φ is called super if the smallest possible labels appear on the vertices. A (super) $(a,d)$ -H-antimagic graph is a graph that admits a (super) $(a,d)$ -H-antimagic total labeling. In this paper the existence of super $(a,d)$ -H-antimagic total labelings for the m-shadow and the closed m-shadow of a connected G for several values of d is proved.