On H-supermagic labelings of m-shadow of paths and cycles

Abstract

A simple graph G = (V, E) is said to be an H-covering if every edge of G belongs to at least one subgraph isomorphic to F. A bijection f: V ∪ E → (1, 2, 3,..., |V| + |E|) is an (a,d)-H-antimagic total labeling of G if, for all subgraphs H isomorphic to H, the sum of labels of all vertices and edges in H form an arithmetic sequence (a, a + d,..., (k − 1)d) where a > 0, d ≥ 0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is called super if the smallest possible labels appear on the vertices. A graph that admits (super) (a, d)-H-antimagic total labeling is called (super) (a, d)-H-antimagic. For a special d = 0, the (super) (a, 0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits such a labeling is called H-(super)magic. The m-shadow of graph G, Dm (G), is a graph obtained by taking m copies of G, namely, G 1 , G 2 ,..., Gm, and then joining every vertex u in Gi, i ∈ (1, 2,..., m − 1), to the neighbors of the corresponding vertex V in Gi +1 . In this paper we studied the H-supermagic labelings of Dm (G) where G are paths and cycles.