An update on the existence of integer Heffter arrays

TL;DR

The paper advances the existence theory for integer Heffter arrays $H(m,n;s;k)$, addressing Archdeacon's conjecture that the natural necessary conditions are also sufficient. It develops new constructions by leveraging base blocks, shiftable arrays, and careful skeleton embeddings to prove existence when $d=\gcd(s,k)\ge5$ with $d\equiv1\pmod4$, and to build large instances from smaller, well-understood blocks. It also provides broad IHS$(m,n;c)$ existence results across multiple parity configurations, using intricate zero-sum block assemblies. Collectively, these results cover all admissible parameter regimes except the narrow gap where $k\in\{3,5\}$ and $\gcd(s,k)=1$ with $s\not\equiv 0\pmod4$, thereby significantly narrowing the open cases and strengthening the conjectured sufficiency of the Archdeacon conditions.

Abstract

An integer Heffter array $H(m,n;s;k)$ is an $m\times n$ partially filled array whose entries are the elements of a subset $Ω\subset \mathbb{Z}$ such that $\{Ω,-Ω\}$ is a partition of the set $\{1,2,\ldots,2nk\}$ and such that the following conditions are satisfied: each row contains $s$ filled cells, each column contains $k$ filled cells, the elements in every row and column add up to $0$. It was conjectured by Dan Archdeacon that an integer $\H(m,n;s;k)$ exists if and only if $ms=nk$, $3\leq s \leq n$, $3\leq k\leq m$ and $nk\equiv 0,3\pmod 4$. In this paper, we provide new constructions of these objects that allow us to prove the validity of Archdeacon's conjecture in each admissible case, except when $k=3,5$ and $s\not \equiv 0\pmod 4$ is such that $\gcd(s,k)=1$.

Figures (1)

• Figure 1: An integer $\mathrm{H}(20,10;9,18)$.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem A
• Theorem B
• Theorem C
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['mainA']}