Let G be a finite undirected simple connected graph with vertex set $V(G)$, distance function ∂ and diameter d. Let $D\subseteq \{0,1,\dots ,d\}$ be a set of distances. A bijection $l:V(G)\to \{1,2,\dots ,|V(G)|\}$ 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 $N_D(v)=\{x\in V(G):\partial (x,v)\in D\}$. We say G has a D-magic labeling if G admits a D-magic labeling. In this paper, we show that the folded n-cube has a {1}-magic labeling (resp. a $\{0,1\}$-magic labeling) if and only if $n\equiv 0$ (mod 4) (resp. $n\equiv 3$ (mod 4)).

中文翻译：

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折叠n -立方体的D -魔法标签

设G是一个有顶点集的有限无向简单连通图$伏(G)$, 距离函数 ∂ 和直径d。让$D\subseteq \{0,1,\dots ,d\}$是一组距离。双射$升：伏(G)\to \{1,2,\dots ,|伏(G)|\}$被称为d的-Magic标记ģ如果存在一个常数ķ使得$\sum _{X\in N_D(v)}升(X)=克$ 对于任何顶点 $v\in 伏(G)$， 在哪里 $N_D(v)=\{X\in 伏(G)：\partial (X,v)\in D\}$. 我们说摹具有d Magic信息标签，如果摹录取d -Magic标签。在本文中，我们展示了折叠的n- cube 具有 {1}-magic 标签（resp. a$\{0,1\}$-magic 标签）当且仅当 $n\equiv 0$ (mod 4) (resp。 $n\equiv 3$ （模式 4））。