Tight globally simple non-zero sum Heffter arrays and biembeddings

Abstract

Square relative non-zero sum Heffter arrays, denoted by $\mathrm{N}\mathrm{H}_t(n;k)$, have been introduced as a variant of the classical concept of Heffter array. An $\mathrm{N}\mathrm{H}_t(n; k)$ is an $n\times n$ partially filled array with elements in $\mathbb{Z}_v$, 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 $x\in \mathbb{Z}_v$ 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 non-zero sum Heffter arrays with no empty cells, $\mathrm{N}\mathrm{H}_t(n;n)$, for every $n$ odd, when $t$ is a divisor of $n$ and when $t\in\{2,2n,n^2,2n^2\}$. 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.