论文标题
存在$λ$ -fold的非零总和HEFFTER ARREARES通过本地注意事项
Existence of $λ$-Fold Non-zero sum Heffter arrays through local considerations
论文作者
论文摘要
在[12]中引入了循环基团，非零和heffter阵列的部分填充阵列类别，这些阵列与heffter阵列一样，与差异族，图分解和biembeddings相关。在这里，我们将此定义概括为任何有限组。给定一个组$ g $的子组$ j $，$ fold non-Zero non-Zero heffter heffter阵列$ g $，相对于$ j $，$ j $，$^λ\ mathrm {n} \ mathrm {h Mathrm {h} _t（m，n; h，k）$，是$ m \ times times n $ \ times n $ p p。 f。带有$ g $的条目的数组：每行包含$ h $填充的单元格，每列包含$ k $填充的单元格；对于g \ setminus j $中的每一个$ x \，$ x $和$ x $的出现总和为$λ$;每个行和列中的元素的总和是按照从左到右的自然顺序进行行，从列的上到底部的自然订购，不同于$ 0 $（$ g $）。在[12]中，如果$λ= 1 $和$ g = \ mathbb {z} _v $，就出现了一个完整的，概率的解决方案，这是本研究的起点。在本文中，我们将考虑到$λ$的通用值和一个通用有限组$ g $的存在问题，我们为此问题提供了几乎完整的解决方案。特别是，我们将通过本地考虑（受洛瓦斯本地引理的启发）证明，如果满足微不足道的必要条件，并且$ | g | = v \ | = v \ geq 41 $，则存在$λ$ fold non-Zero heffter heffter and heffter and heffter阵列。如果数组不包含空单元格，则可以将此值降至$ 29 $。最后，我们将证明这些阵列会导致多编码的生物贴花到可定向的表面，并提供此类嵌入的新型无限家庭。
In [12] was introduced, for cyclic groups, the class of partially filled arrays of the non-zero sum Heffter array that are, as the Heffter arrays, related to difference families, graph decompositions, and biembeddings. Here we generalize this definition to any finite groups. Given a subgroup $J$ of order $t$ of a group $G$, a $λ$-fold non-zero sum Heffter array over $G$ relative to $J$, $^λ\mathrm{N}\mathrm{H}_t(m,n; h,k)$, is an $m \times n$ p. f. array with entries in $G$ such that: each row contains $h$ filled cells and each column contains $k$ filled cells; for every $x\in G\setminus J$, the sum of the occurrence of $x$ and $-x$ is $λ$; the sum of the elements in every row and column is, following the natural orderings from left to right for the rows and from top to bottom for the columns, different from $0$ (in $G$). In [12], there was presented a complete, probabilistic, solution for the existence problem in case $λ=1$ and $G=\mathbb{Z}_v$ that is the starting point of this investigation. In this paper, we will consider the existence problem for a generic value of $λ$ and a generic finite group $G$, and we present an almost complete solution to this problem. In particular, we will prove, through local considerations (inspired by Lovász Local Lemma), that there exists a $λ$-fold non-zero sum Heffter array over $G$ relative to $J$ whenever the trivial necessary conditions are satisfied and $|G|=v\geq 41$. This value can be turned down to $29$ in case the array does not contain empty cells. Finally, we will show that these arrays give rise to biembeddings of multigraphs into orientable surfaces and we provide new infinite families of such embeddings.
