A Partial Solution to an Open Problem on Super Edge-Magic Total Labelings of (4,3)-Cycle Books Graphs

Let H be a graph with h vertices and e edges. An edge-magic-total labeling of graph H is a bijective function g : V(H) boolean OR E(H)->{1, 2, & mldr;, h + e} such that g(a) + g(a, b) + g(b) = k, for all(a, b) is an element of E(H). When g : V(H)->{1, 2, & mldr;, h}, it is termed a super edge-magic total labeling. We introduce the concept of a (c, d)-cycle book graph, denoted by Book[(c, alpha), (d, beta), 2], constructed from alpha copies of cycles of order c and beta copies of cycles of order d, each sharing a common path P2. The super edge-magic total labelings problem of Book[(c, alpha), (d, beta), 2] remains unsolved. In this study, we contribute a partial solution to the super edge-magic total labelings of Book[(4, alpha), (3, beta), 2]. Specifically, we establish that the graph Book[(4, alpha), (3, beta), 2] admits a super edge-magic total labeling for all alpha, where alpha is an even positive integer and beta = 1, 2. Using the super edge-magic total labelings of Book[(4, 26), (3, 2), 2], we give an illustration how to encrypt and decrypt a secret message in cryptography by the Rivest, Shamir, and Adleman (RSA) algorithm.