The existence of square non-integer Heffter arrays

TL;DR

This paper proves the existence of non-integer square Heffter arrays for all parameters where the array size exceeds the number of filled cells per row and column, expanding the known existence results.

Contribution

It establishes the existence of non-integer square Heffter arrays for all cases with 3 ≤ k < n, generalizing previous integer-only results.

Findings

Non-integer square Heffter arrays exist for all n and k with 3 ≤ k < n.

This extends the known existence conditions beyond integer arrays.

Provides new constructions for non-integer Heffter arrays.

Abstract

A Heffter array $H(n;k)$ is an $n\times n$ matrix such that each row and column contains $k$ filled cells, each row and column sum is divisible by $2nk+1$ and either $x$ or $-x$ appears in the array for each integer $1\leq x\leq nk$. Heffter arrays are useful for embedding the graph $K_{2nk+1}$ on an orientable surface. An integer Heffter array is one in which each row and column sum is $0$. Necessary and sufficient conditions (on $n$ and $k$) for the existence of an integer Heffter array $H(n;k)$ were verified by Archdeacon, Dinitz, Donovan and Yaz\i c\i \ (2015) and Dinitz and Wanless (2017). In this paper we consider square Heffter arrays that are not necessarily integer. We show that such Heffter arrays exist whenever $3\leq k<n$.