On the existence of integer relative Heffter arrays

TL;DR

This paper investigates the existence conditions for integer relative Heffter arrays with even parameters, providing new existence results based on divisibility and parity conditions of array dimensions and parameters.

Contribution

It establishes new existence theorems for integer relative Heffter arrays when both row and column parameters are even, depending on their congruence classes modulo 4.

Findings

Existence when s,k are multiples of 4 for given parameters.

Existence when s ≡ 2 mod 4 and k ≡ 0 mod 4, with m even.

Existence when s ≡ 0 mod 4 and k ≡ 2 mod 4, with n even.

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 $\pm\left\{ 1, \ldots, \left\lfloor \frac{v}{2}\right\rfloor \right\}$, 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 $4\leq s\leq n$ and $4\leq k \leq m$ are such that $ms=nk$. Let $t$ be a divisor of $2ms$. (a) If $s,k \equiv 0 \pmod 4$, there exists an integer $H_t(m,n;s,k)$. (b) If $s\equiv 2\pmod 4$ and $k\equiv 0 \pmod 4$, there exists an integer $H_t(m,n;s,k)$ if and only if $m$ is even. (c) If $s\equiv 0\pmod 4$ and $k\equiv 2 \pmod 4$, 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 $s,k\equiv 2 \pmod 4$, then there exists an integer $H_t(m,n;s,k)$.