AI 中文总结
本文利用组合零点定理，对简单图\(G\)，在给定整数\(k\geq 2\)下，研究边标记\(h\)诱导顶点标记的情况，分析\(G\)成为\(t -\)和\(\mathbb Z_k -\)魔术图以及\(\mathbb{Z}_{k}-\)反魔术图的条件。
AI 中文摘要
设\(G=(V,E)\)为简单图，\(k\geq 2\)为整数。对于边标记\(h:E(G)\to \mathbb{Z}_{k} \backslash \{0\}\)，通过\(h^+(v)=\sum_{e \ni v} h(e) \pmod{k}\)定义诱导顶点标记。对于\(t\in \mathbb Z_k\)，若存在这样的标记\(h\)使\(h^+(v)=t\)对所有\(v\in V\)成立，则称\(G\)是\(t -\)和\(\mathbb Z_k -\)魔术的；若对某个\(t\)成立，则称\(G\)是\(\mathbb Z_k -\)魔术的。类似地，若存在边标记\(h\)使诱导顶点标记\(h^+(v)\)是单射，则称\(G\)是\(\mathbb{Z}_{k}-\)反魔术的。本文用组合零点定理分析这两种魔术图标记。
英文摘要
Let $G=(V,E)$ be a simple graph, and let $k\geq 2$ be an integer. For an edge labeling $h:E(G)\to \mathbb{Z}_{k} \backslash \{0\}$, define the induced vertex label by \[ h^+(v)=\sum_{e \ni v} h(e) \pmod{k}. \] For $t\in \mathbb Z_k$, we say that $G$ is \emph{$t$-sum $\mathbb Z_k$-magic} if there exists such a labeling $h$ satisfying \[ h^+(v)=t \qquad\text{for all }v\in V. \] We say that $G$ is \emph{$\mathbb Z_k$-magic} if $G$ is $t$-sum $\mathbb Z_k$-magic for some $t\in \mathbb Z_k$. Similarly, if there exists an edge labeling $h: E(G) \to \mathbb{Z}_{k} \backslash \{0\}$ such that the induced vertex labeling $h^+(v)=\sum_{e\ni v} h(e)$ (mod $k$) is injective, then $G$ is called \emph{$\mathbb{Z}_{k}$-antimagic}. In this paper, we use the Combinatorial Nullstellensatz to analyze these two types of magic graph labelings.
Comments15 pages, 2 figures
