Note on Super (A,1)–P3–Antimagic Total Labeling of Star Sn

Abstract

Let G=(V,E) be a simple graph and H be a subgraph of G. Then 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 bijection f:V(G)∪E(G)→{1,2,3,…,|V(G)|+|E(G)|} such that for all subgraphs H′ of G isomorphic to H, the H′ weights w(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(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. The labeling f is called a super (a,d)−H-antimagic total labeling if f(V(G))={1,2,3,…,|V(G)|}. In [5], David Laurence and Kathiresan posed a problem that characterizes the super (a,1)−P3-antimagic total labeling of Star Sn, where n=6,7,8,9. In this paper, we completely solved this problem.