On the existence of integer relative Heffter arrays

Abstract

Let $v=2ms+t$ be a positive integer, where $t$ divides $2ms$, and let $J$ be the subgroup of order $t$ of the cyclic group ${\mathbb{Z}}_v$. An integer Heffter array $H_t(m,n;s,k)$ over ${\mathbb{Z}}_v$ relative to $J$ is an $m\times n$ partially filled array with elements in ${\mathbb{Z}}_v$ such that: (a) each row contains $s$ filled cells and each column contains $k$ filled cells; (b) for every $x\in {\mathbb{Z}}_v\setminus J$, either $x$ or $-x$ appears in the array; (c) the elements in every row and column, viewed as integers in $\pm \left\{1,\dots ,\lfloor \frac{v}{2}\rfloor \right\}$, sum to $0$ in $\mathbb{Z}$. In this paper we study the existence of an integer $H_t(m,n;s,k)$ when $s$ and $k$ are both even, proving the following results. Suppose that $4\le s\le n$ and $4\le k\le m$ are such that $ms=nk$. Let $t$ be a divisor of $2ms$. (a) If $s,k\equiv 0(\mathrm{mod}4)$, there exists an integer $H_t(m,n;s,k)$. (b) If $s\equiv 2(\mathrm{mod}4)$ and $k\equiv 0(\mathrm{mod}4)$, there exists an integer $H_t(m,n;s,k)$ if and only if $m$ is even. (c) If $s\equiv 0(\mathrm{mod}4)$ and $k\equiv 2(\mathrm{mod}4)$, then there exists an integer $H_t(m,n;s,k)$ if and only if $n$ is even. (d) Suppose that $m$ and $n$ are both even. If $s,k\equiv 2(\mathrm{mod}4)$, then there exists an integer $H_t(m,n;s,k)$.

In this version, we also construct non-square relative Heffter arrays