Book graphs are cycle antimagic

Let G = ( V , E ) be a finite simple graph with v = | V ( G ) | vertices and e = | E ( G ) | edges. Further suppose that H := { H 1 , H 2 , … , H t } is a family of subgraphs of G . In case, each edge of E ( G ) belongs to at least one of the subgraphs H i from the family H , we say G admits an edge-covering. When every subgraph H i in H 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 η : V ∪ E → { 1 , 2 , … , v + e } such that for each subgraph H ′ of G isomorphic to H , the sum of labels of all the edges and vertices belongs to H ′ constitutes an arithmetic progression with the initial term a and the common difference d . For η ( V ) = { 1 , 2 , 3 , … , v } , the graph G is said to be super ( a , d ) − H -antimagic and for d = 0 it is called H -supermagic. When the given graph H is a cycle C m then H -covering is called C m -covering and super ( a , d ) − H -antimagic labeling becomes super ( a , d ) − C m -antimagic labeling. In this paper, we investigate the existence of super ( a , d ) − C m -antimagic labeling of book graphs B n , for m = 4 , n ≥ 2 and for differences d = 1 , 2 , 3 , … , 13 .