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, . . . , $\mid V (G) \cup E(G)\mid$ with the property that, given any vertex $x, \lambda(x) +\Sigma_y\sim x \lambda(y) = k$ for some constant $k$. In this paper we completely determine which complete bipartite graphs have vertex-magic total labelings.