On super (a, d)-h-antimagic total covering of star related graphs

Abstract

Let G = (V (G),E(G)) be a simple graph and H be a subgraph of G. G admits an H-covering, if every edge in E(G) belongs to at least one subgraph of G that is isomorphic to H. An (a, d)-H-antimagic total labeling of G is a bijection λ : V (G) ? E(G) → {1, 2, 3, . . . , |V (G)| + |E(G)|} such that for all subgraphs H? isomorphic to H, the H? weights wt(H?)=XvεV(H?) λ(v)+XeεE(H?) λ(e) constitute an arithmetic progression a, a+d, a+2d, . . . , a+(n-1)d where a and d are positive integers and n is the number of subgraphs of G isomorphic to H. Additionally, the labeling λ is called a super (a, d)-H-antimagic total labeling if λ(V (G)) = {1, 2, 3, . . . , |V (G)|}. In this paper we study super (a, d)-H-antimagic total labelings of star related graphs Gu[Sn] and caterpillars.