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

An [Formula: see text]-H-antimagic total labeling of a simple graph G admitting an H-covering is a bijection [Formula: see text] such that for all subgraphs [Formula: see text] of G isomorphic to H, the set of [Formula: see text]-weights given by [Formula: see text] forms an arithmetic sequence [Formula: see text] where [Formula: see text] , [Formula: see text] 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) [Formula: see text]-H-antimagic graph is a graph that admits a (super) [Formula: see text]-H-antimagic total labeling. In this paper the existence of super [Formula: see text]-H-antimagic total labelings for the m-shadow and the closed m-shadow of a connected G for several values of d is proved.