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

Abstract

An (a,d)-H-antimagic total labeling of a simple graph G admitting an H-covering is a bijection φ:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that for all subgraphs H′ of G isomorphic to H, the set of H′-weights given by wtφ(H′)=∑v∈V(H′)φ(v)+∑e∈E(H′)φ(e) forms an arithmetic sequence a,a+d,…,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.