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) ∪ E(H)⟶{1, 2, …, h + e} such that g(a) + g(a, b) + g(b) = k, ∀(a, b) ∈ E(H). When g : V(H)⟶{1, 2, …, 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, α), (d, β), 2], constructed from α copies of cycles of order c and β copies of cycles of order d, each sharing a common path P2. The super edge‐magic total labelings problem of Book[(c, α), (d, β), 2] remains unsolved. In this study, we contribute a partial solution to the super edge‐magic total labelings of Book[(4, α), (3, β), 2]. Specifically, we establish that the graph Book[(4, α), (3, β), 2] admits a super edge‐magic total labeling for all α, where α is an even positive integer and β = 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.