Vertex Magic Total Labellings of Complete Multipartite Graphs

Abstract

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 $\{1,2,\dots ,e+v\}$ 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 $n\equiv 3(\mathrm{mod}4)$.