Super Edge-Magic Labelings of Book Graphs \(B_n\)

Abstract

A graph \(G\) is called super edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv) = C\) is a constant for any \(uv \in E(G)\) and \(f(V(G)) = \{1, 2, \ldots, |V(G)|\}\), \(f(E(G)) = \{|V(G)| + 1, |V(G)| + 2, \ldots, |V(G)| + |E(G)|\}\). R. M. Figueroa-Centeno et al. provided the following conjecture: For every integer \(n \geq 5\), the book \(B_n\) is super edge-magic if and only if \(n\) is even or \(n \equiv 5 \pmod 8\). In this paper, we show that \(B_n\) is super edge-magic for even \(n \geq 6\).