Rectangular Heffter arrays: a reduction theorem

TL;DR

This paper introduces a reduction method to construct rectangular Heffter arrays from square arrays, providing existence results for various parameter cases and extending to signed magic arrays and magic rectangles.

Contribution

It presents a novel reduction theorem that constructs rectangular Heffter arrays from square arrays, expanding the known existence cases and applying to related array types.

Findings

Existence of Heffter arrays in multiple parameter cases.

Construction methods for signed magic arrays and magic rectangles.

New existence results when certain divisibility conditions are met.

Abstract

Let $m,n,s,k$ be four integers such that $3\leq s \leq n$, $3\leq k\leq m$ and $ms=nk$. Set $d=\gcd(s,k)$. In this paper we show how one can construct a Heffter array $H(m,n;s,k)$ starting from a square Heffter array $H(nk/d;d)$ whose elements belong to $d$ consecutive diagonals. As an example of application of this method, we prove that there exists an integer $H(m,n;s,k)$ in each of the following cases: $(i)$ $d\equiv 0 \pmod 4$; $(ii)$ $5\leq d\equiv 1 \pmod 4$ and $n k\equiv 3\pmod 4$; $(iii)$ $d\equiv 2 \pmod 4$ and $nk\equiv 0 \pmod 4$; $(iv)$ $d\equiv 3 \pmod 4$ and $n k\equiv 0,3\pmod 4$. The same method can be applied also for signed magic arrays $SMA(m,n;s,k)$ and for magic rectangles $MR(m,n;s,k)$. In fact, we prove that there exists an $SMA(m,n;s,k)$ when $d\geq 2$, and there exists an $MR(m,n;s,k)$ when either $d\geq 2$ is even or $d\geq 3$ and $nk$ are odd. We also provide constructions of integer Heffter arrays and signed magic arrays when $k$ is odd and $s\equiv 0 \pmod 4$.