\(D\)-magic labelings of the folded \(n\)-cube
Let $G$ be a finite undirected simple connected graph with vertex set $V(G)$, distance function $\delta$ and diameter $d$. Let $D\subset\{0,1,\dots,d\}$ be a set of distances. A bijection $l:V(G)\to\{0,1,\dots,|V(G)|\}$ is called a $D$-magic labeling of $G$ if there exists a constant $k$ such that $\sum_{x\in N_D(V)}l(x)=k$ for any vertex $v\in V(G)$, where $N_D(v)=\{x\in V(G):\delta(x,v)\in D\}$. $G$ has a $D$-magic labeling if $G$ admits a $D$-magic labeling. In this paper, the authors prove that the folded $n$-cube has a $\{1\}$-magic if and only if $n\equiv 0\pmod4$.
scientific article; zbMATH DE number 7369648