Communications of the
Korean Mathematical Society CKMS

An $H$-magic labeling in a $H$-decomposable graph $G$ is a bijection $ f: V(G)\cup E(G)\rightarrow {\{1,2, \ldots ,p+q\}}$ such that for every copy $H$ in the decomposition, $ \sum_{v\in V(H)} f(v)+\sum_{e\in E(H)} f(e)$ is constant. $f$ is said to be $H$-$V$-super magic if $f(V(G))={\{1,2, \ldots ,p\}}$. In this paper, we prove that complete bipartite graphs $K_{n,n}$ are $H$-$V$-super magic decomposable where $ H\cong K_{1,n} $ with $n\geq1$.

Keywords: $H$-decomposable graph, $H$-$V$-super magic labeling, complete bipartite graph

MSC numbers: $05C78$, $05C70$