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On H -Supermagic Labelings of m -Shadow of Paths and Cycles

2019; Hindawi Publishing Corporation; Volume: 2019; Linguagem: Inglês

10.1155/2019/8780329

1687-0425

Ika Hesti Agustin, Faisal Susanto, Dafik Dafik, Rafiantika Megahnia Prihandini, Ridho Alfarisi, I Wayan Sudarsana,

graph theory and CDMA systems

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 H . 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 , D m ( G ) , is a graph obtained by taking m copies of G , namely, G 1 , G 2 , … , G m , and then joining every vertex u in G i , i ∈ { 1,2 , … , m - 1 } , to the neighbors of the corresponding vertex v in G i + 1 . In this paper we studied the H -supermagic labelings of D m ( G ) where G are paths and cycles.