Super ( a , d ) - C_4 -antimagicness of book graphs

Let G = ( V , E ) be a~finite simple graph with | V ( G ) | vertices and | E ( G ) | edges. An edge-covering of G is a family of subgraphs H 1 , H 2 , … , H t such that each edge of E ( G ) belongs to at least one of the subgraphs H i , i = 1 , 2 , … , t . If every subgraph H i is isomorphic to a given graph H , then the graph G admits an H -covering. A graph G admitting H covering is called an ( a , d ) - H -antimagic if there is a bijection f : V ∪ E → { 1 , 2 , … , | V ( G ) | + | E ( G ) | } such that for each subgraph H ′ of G isomorphic to H , the sum of labels of all the edges and vertices belonged to H ′ constitutes an arithmetic progression with the initial term a and the common difference d . For f ( V ) = { 1 , 2 , 3 , … , | V ( G ) | } , the graph G is said to be super ( a , d ) - H -antimagic and for d = 0 it is called H -supermagic. In this paper, we investigate the existence of super ( a , d ) - C 4 -antimagic labeling of book graphs, for difference d = 0 , 1 and n ≥ 2 .