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

Abstract

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 uεV(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 Ca and n copies cycle Cb with a commont path Pt . 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.