On Edge Magic Total Labeling of (7, 3)-Cycle Books

Abstract

A graph G is called (a, b)-cycle books B[(Ca, m), (Cb, n), Pt] if G consists of m cycles Ca and n cycles Cb with a common path Pt. In this article we show that the (7, 3)-cycle books B[(C7, 1), (C3, n), P2] admits edge-magic total labeling. In addition we prove that (a, 3)-cycle books B[(Ca, 1), (C3, n), P2] admits super edge-magic total labeling for a = 7 and extends the values of a to 4x − 1 for any positive integer x. Moreover we prove that the (7, 3)-cycle books B[(C7, 2), (C3, n), P2] admits super edge-magic total labeling.

A Conjecture on Super Edge-Magic Total Labeling of 4-Cycle Books

“…Let m ≥ 1 be any positive integer and let C a be the cycles of order a. Graph B[(a, m), 2] is constructed from some cycles C a of the same order. Swita et al [9] contructed a graph from some cycles with different orders. A graph (a, b) − cycle book B[(a, m), (b, n), t] is constructed from some cycles C a and C b with a common path P t , a path of order t with m, n, a, b, and t as the positive integers.…”

“…e second reason is both Problems 1 and 2 provide a challenging problem for the researchers, since they are open problems. Swita et al [9] proved Problem 2 for a � 7 or a � 4x − 1 for any integer x. MacDougall and Wallis [13]…”