A Note On Magic Graphs

By a \((1,1)\) edge-magic labeling of a graph \( G(V, E) \), we mean a bijection \( f \) from \( V \cup E \) to \(\{1, \dots, |V| + |E|\}\) such that for all edges \( uv \in E(G) \), the value \( f(u) + f(v) + f(uv) \) is constant. We provide a different proof of a well-known result in additive number theory by Paul Erdős and, interestingly, demonstrate a practical application of this result. Additionally, we make some progress using computational methods towards the conjecture proposed by Yegnanarayanan: “Every graph on \( p \geq 9 \) vertices can be embedded as a subgraph of some \((1,1)\) edge-magic graph.”