论文标题
阿贝尔组的零和分区及其应用于魔术和反象征类型的标签
Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings
论文作者
论文摘要
自80年代以来，已经知道以下问题。令$γ$为ABELIAN订单$ m $（表示$ |γ| = m $），让$ t $和$ \ {m_i \} _ {i = 1}^{t} $，成为积极的整数，以便$ \ sum__ = 1}^i = 1}^t m_i = m_i = m-1 $。确定$γ^*=γ\ setMinus \ {0 \} $，可以将$γ$的非零元素集划分为偏置子集$ \ {s_i \} _ {i = 1} $ 1 \ leq i \ leq t $。这样的子集分区称为\ textit {零和分区}。 $ | i（γ）| \ neq 1 $，其中$ i（γ）$是$γ$中的一组，是存在零和分区的必要条件。在本文中，我们表明每$ 1 \ leq i \ leq t $的$ M_I \ geq 4 $的附加条件就足够了。此外，我们将零和分区的某些应用与图形的魔术和反象征类型标记相关。
The following problem has been known since the 80s. Let $Γ$ be an Abelian group of order $m$ (denoted $|Γ|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $Γ^*=Γ\setminus\{0\}$, the set of non-zero elements of $Γ$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(Γ)|\neq 1$, where $I(Γ)$ is the set of involutions in $Γ$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.
