Tight globally simple nonzero sum Heffter arrays and biembeddings

Square relative nonzero sum Heffter arrays, denoted by $N{H}_t\left(n;k\right)$, have been introduced as a variant of the classical concept of Heffter array. An $N{H}_t\left(n;k\right)$ is an $n\times n$ partially filled array with elements in $Z_v$, where $v=2nk+t$, whose rows and whose columns contain $k$ filled cells, such that the sum of the elements in every row and column is different from 0 (modulo $v$) and, for every $x\in Z_v$ not belonging to the subgroup of order $t$, either $x$ or $-x$ appears in the array. In this paper we give direct constructions of square nonzero sum Heffter arrays with no empty cells, $N{H}_t\left(n;n\right)$, for every $n$ odd, when $t$ is a divisor of $n$ and when $t\in \left\{2,2n,n^2,2n^2\right\}$. The constructed arrays have also the very restrictive property of being “globally simple”; this allows us to get new orthogonal path decompositions and new biembeddings of complete multipartite graphs.