A case study of an edge-magic total labeling of (a,b)-cycle books

Suppose α and β be the order and size of a graph G respectively. A one-one function h which maps the set of vertices and edges of a graph G onto the integers 1, 2, 3, …, α + β such that h ( u ) + h ( uv ) + h ( v ) = c for any edge ( uv ) ε E ( G ) is called an edge-magic total labeling of If h ( u ) ε {1,2, …, α} for any u ε V ( G ) then h is a super edge-magic total labeling of G . One of interesting research topic is super edge-magic total labeling of cycle book. A cycle book B ( a, m, b, n, t ) is made up from m copies of cycle C a and n copies cycle C b with a commont path P t . Super edge-magic total labeling of a cycle book B ( a, m, b, n , 2) is still under investigation even for the case a = b This paper talk about a partial solution to this problem. We prove that a cycle book B (5, 2, 3, n , 2) has a super edge-magic total labeling for all positive integr n . In addition, we show that a cycle book B (5, 2, 3, n, 2) have an edge-magic total labeling for an integer n , 1 ≥ n ≥ 6.