两个圈的卡氏积的群超幻标号

Group Supermagic Labeling of the Cartesian Product of Two Cycles

摘要: 图的各种幻型标号的存在性是近年来图论中的一个研究热点。在该领域的研究中用到了图论、群论、数论、线性代数等数学工具。设$G=\left(V,E\right)$ 是一个有限简单图，$A$ 是一个有限交换群。$G$ 的一个$A$ -超幻标号是一个双射$\phi :E\to A$ ，满足存在$c\in A$ ，使得对所有的$x\in V$ ，有${\displaystyle {\sum }_{e\in E\left(x\right)}\phi \left(e\right)}=c$ ，其中$E\left(x\right)$ 代表所有与顶点$x$ 关联的边的集合。对任意$n\ge 3$ ，记$C_n$ 是$n$ 个顶点的圈。我们利用有限交换群元素的排列得到了$C_m\square C_n$ 上$A$ -超幻标号存在性的新的充分条件。

Abstract: The existence of various magic-type labeling of graphs is a research hotspot in recent years. Many mathematical tools, such as graph theory, group theory, number theory, and linear algebra, are used in the research of this area. Let $G=\left(V,E\right)$ be a finite simple graph and let $A$ be a finite abelian group. An $A$ -supermagic labeling of $G$ is a bijection $\phi :E\to A$ which satisfies that there exists $c\in A$ , such that ${\displaystyle {\sum }_{e\in E\left(x\right)}\phi \left(e\right)}=c$ for any $x\in V$ , where $E\left(x\right)$ is the set of edges which adjacent to $x$ . For any $n\ge 3$ , let $C_n$ denote the cycle with $n$ vertices. We use some results on the arrangement of elements of finite abelian groups to obtain new sufficient conditions such that the Cartesian product of cycles $C_m\square C_n$ admits $A$ -supermagic labeling.