The existence of square 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 $1\le x\le nk$. 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 Yazıcı\ (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 $3\le k<n$.