Vertex Magic Total Labellings of Complete Multipartite Graphs

A Vertex Magic Total Labeling of a graph \( G \) is a one-to-one map \( \lambda \) from \( E(G) \cup V(G) \) onto the set of integers \( \{1, 2, \ldots, e + v\} \) such that for all \( x \in V \) we have

\[
\lambda(x) + \sum \lambda(xy) = h
\]

for some constant \( h \), where the sum is taken over all vertices \( y \) adjacent to \( x \). In this paper, we present several theorems on the existence of such labelings for multipartite graphs and give constructions for labelings for two infinite families of complete tripartite graphs, namely \( K_{1,n,n} \) for odd \( n \) and \( K_{2,n,n} \) for \( n \equiv 3 \pmod{4} \).