A generalization of Heffter arrays

In this paper, we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let $v=2nk+t$ be a positive integer, where $t$ divides $2nk$, and let $J$ be the subgroup of ${\mathbb{Z}}_v$ of order $t$. A ${\mathrm{H}}_t\left(m,n;s,k\right)$ Heffter array 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 $x\in {\mathbb{Z}}_v\backslash J$, either $x$ or $?x$ appears in the array; and (c) the elements in every row and column sum to $0$. Here we study the existence of square integer (i.e., with entries chosen in $\pm \left\{1,\text{…},?\frac{2nk+t}{2}?\right\}$ and where the sums are zero in $\mathbb{Z}$) relative Heffter arrays for $t=k$, denoted by ${\mathrm{H}}_k\left(n;k\right)$. In particular, we prove that for $3\le k\le n$, with $k\ne 5$, there exists an integer ${\mathrm{H}}_k\left(n;k\right)$ if and only if one of the following holds: (a) $k$ is odd and $n\equiv 0,3\left(\text{mod}4\right)$; (b) $k\equiv 2\left(\text{mod}4\right)$ and $n$ is even; (c) $k\equiv 0\left(\text{mod}4\right)$. Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.