Towards a solution of Archdeacon's conjecture on integer Heffter arrays

TL;DR

This paper advances the understanding of Archdeacon's conjecture by proving the existence of integer Heffter arrays under certain conditions, using new constructions to confirm the conjecture in specific cases.

Contribution

It provides a proof of Archdeacon's conjecture for a broad class of integer Heffter arrays when certain divisibility and parity conditions are met, through novel array constructions.

Findings

Confirmed the conjecture for arrays with k ≥ 7·gcd(s,k) when this is odd

Constructed integer Heffter array sets satisfying the conjecture

Extended the known cases where the conjecture holds

Abstract

In this paper, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays $H(m,n;s,k)$ whenever the necessary conditions hold, that is, $3\leqslant s \leqslant n$, $3\leqslant k\leqslant m$, $ms=nk$ and $nk\equiv 0,3 \pmod 4$. By constructing integer Heffter array sets, we prove the conjecture in the affirmative whenever $k\geqslant 7\cdot \gcd(s,k)$ is odd and $s\neq 3,5,6,10$.