Square Integer Heffter Arrays with Empty Cells

TL;DR

This paper studies square Heffter arrays with empty cells, focusing on their construction where each row and column sum to zero, and provides solutions for most cases.

Contribution

It extends the theory of Heffter arrays by constructing square arrays with empty cells that maintain zero sums, filling a gap in existing research.

Findings

Most instances of square Heffter arrays with empty cells are solved.

Constructed arrays maintain zero row and column sums.

Results facilitate embedding complete graphs on surfaces.

Abstract

A Heffter array $H(m,n;s,t)$ is an $m \times n$ matrix with nonzero entries from $\mathbb{Z}_{2ms+1}$ such that $i)$ each row contains $s$ filled cells and each column contains $t$ filled cells, $ii)$ every row and column sum to 0, and $iii)$ no element from $\{x,-x\}$ appears twice. Heffter arrays are useful in embedding the complete graph $K_{2nm+1}$ on an orientable surface where the embedding has the property that each edge borders exactly one $s-$cycle and one $t-$cycle. Archdeacon, Boothby and Dinitz proved that these arrays can be constructed in the case when $s=m$, i.e. every cell is filled. In this paper we concentrate on square arrays with empty cells where every row sum and every column sum is $0$ in $\mathbb{Z}$. We solve most of the instances of this case.