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arXiv 1204.0705math.CO
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关于群距离幻图$G[C_4]$的注记

Note on group distance magic graphs $G[C_4]$

Sylwia Cichacz

发表信息：
Graph and Combinatorics 2013
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摘要：

图$G(V,E)$与$|V | = n$的\emph{group distance magic labeling}或$\gr$ -距离魔术标记是从$V$到阶为$n$的阿贝尔群$\gr$的注入$f$，使得每个顶点$x \in V$的权重$w(x)=\sum_{y\in N_G(x)}f(y)$等于相同的元素$μ\in \gr$，称为魔术常数。本文证明了：如果$G$是一个阶为$n=2^{p}(2k+1)$的图，对于某些自然数$p$，$k$，使得对于任何$v\in V(G)$，对于某些常数$c$，$°(v)\equiv c \imod {2^{p+1}}$，则对于图$G[C_4]$的任何阿贝尔群$\gr$，存在$\gr$ -距离魔标号.进一步证明了：如果$\gr$是任意的$4n$阶阿贝尔群，使得对某个$n$阶阿贝尔群$\gA$，有$\gr \cong \zet_2 \times\zet_2 \times \gA$，则对任意图$G[C_4]$，存在$\gr$ -距离魔标号.

英文摘要：

A \emph{group distance magic labeling} or a $\gr$-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group $\gr$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same element $μ\in \gr$, called the magic constant. In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that $°(v)\equiv c \imod {2^{p+1}}$ for some constant $c$ for any $v\in V(G)$, then there exists an $\gr$-distance magic labeling for any Abelian group $\gr$ for the graph $G[C_4]$. Moreover we prove that if $\gr$ is an arbitrary Abelian group of order $4n$ such that $\gr \cong \zet_2 \times\zet_2 \times \gA$ for some Abelian group $\gA$ of order $n$, then exists a $\gr$-distance magic labeling for any graph $G[C_4]$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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