Vertex-Magic Total Labelings of Complete Bipartite Graphs

Abstract

A vertex-magic total labeling on a graph \(G\) is a one-to-one map \(\lambda\) from \(V(G) \cup E(G)\) onto the integers \(1, 2, \ldots, |V(G) \cup E(G)|\) with the property that, given any vertex \(x\), \(\lambda(x) + \sum_{y \sim x} \lambda(y) = k\) for some constant \(k\).

In this paper, we completely determine which complete bipartite graphs have vertex-magic total labelings.