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:VH∪EH⟶1,2,…,h+e such that ga+ga,b+gb=k,∀a,b∈EH. When g:VH⟶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 Bookc,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 Bookc,α,d,β,2 remains unsolved. In this study, we contribute a partial solution to the super edge-magic total labelings of Book4,α,3,β,2. Specifically, we establish that the graph Book4,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 Book4, 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.