Existence of $\lambda$-Fold Non-zero sum Heffter arrays through local considerations

TL;DR

This paper extends the concept of non-zero sum Heffter arrays to arbitrary finite groups, proving their existence under certain conditions using probabilistic methods, and explores their applications in graph embeddings.

Contribution

It generalizes the existence results of $ ext{N} ext{H}_t(m,n; h,k)$ arrays to all finite groups for any $ ext{lambda}$, using local probabilistic considerations.

Findings

Existence of $ ext{N} ext{H}_t(m,n; h,k)$ arrays for groups of order at least 41.

Reduced the order requirement to 29 for arrays without empty cells.

Constructed new infinite families of biembeddings of multigraphs into orientable surfaces.

Abstract

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 $\lambda$-fold non-zero sum Heffter array over $G$ relative to $J$, $^\lambda\mathrm{N}\mathrm{H}_t(m,n; h,k)$, 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 $\lambda$; 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 $\lambda=1$ 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 $\lambda$ 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 Lov\'asz Local Lemma), that there exists a $\lambda$-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.