Construction of Super H-Antimagicness of Graph by Uses a Partition Technique with Cancelation Number

Abstract

Abstract—The graph operation is one method to construct a new graph by applying the operation to two or more graph. One of graph operation is amalgamation, let {Hi} be a finite collection of nontrivial, simple and undirected graphs and let each Hi has a fixed vertex vj called a terminal. The terminal of graph operation is formed by taking all the Hi’s and identifying their terminal. When Hi are all isomorphic graphs, for any positif integer n, we denote such amalgamation by G = Amal(H, v, n), where n denotes the number of copies of H and v is the terminal. The graph G is said to be an (a, d)-H-antimagic total graph if there exist a bijective function f : V (G) ∪ E(G) → {1, 2, . . . , |V (G)| + |E(G)|} such that for all subgraphs isomorphic to H, the total H-weights W(H) = ∑v∈V (H) f(v) + ∑e∈E(H) f(e) form an arithmetic sequence {a, a + d, a + 2d, ..., a + (n − 1)d}, where a and d are positive integers and n is the number of all subgraphs isomorphic to H. An (a, d)-H-antimagic total labeling f is called super if the smallest labels appear in the vertices. In this paper, we study a super (a, d)-H antimagic total labeling of connected of graph G = Amal(H, Ps+2, n) by uses a partition technique with cancelation number. The result is graph G = Amal(H, Ps+2, n) admits a super(a, d)-H antimagic total labeling for almost feasible difference d.