Weak Heffter Arrays and biembedding graphs on non-orientable surfaces

TL;DR

This paper introduces weak Heffter arrays, explores their fundamental properties, and investigates their connections to biembeddings on non-orientable surfaces, filling a gap in the literature.

Contribution

It is the first study on weak Heffter arrays, providing necessary conditions, existence and non-existence results, and linking them to biembeddings on non-orientable surfaces.

Findings

Established necessary conditions for weak Heffter arrays.

Proved existence and non-existence results for certain parameters.

Connected weak Heffter arrays to biembeddings in non-orientable surfaces.

Abstract

In 2015, Archdeacon proposed the notion of Heffter arrays in view of its connection to several other combinatorial objects. In the same paper he also presented the following variant. A weak Heffter array $\mathrm{W}\mathrm{H}(m,n;h,k)$ is an $m \times n$ matrix $A$ such that: each row contains $h$ filled cells and each column contains $k$ filled cells; for every $x \in \mathbb{Z}_{2nk+1} \setminus \{0\}$, there is exactly one cell of $A$ whose element is one of the following: $x,-x,\pm x,\mp x$, where the upper sign on $\pm$ or $\mp$ is the row sign and the lower sign is the column sign; the elements in every row and column (with the corresponding sign) sum to $0$ in $\mathbb{Z}_{2nk+1}$. Also the ``weak concept'', as the classical one, is related to several other topics, such as difference families, cycle systems and biembeddings. Many papers on Heffter arrays have been published, while no one on weak Heffter arrays has been written. This is the first one and here we explore necessary conditions, existence and non-existence results, and connections to biembeddings into non-orientable surfaces.