Tight globally simple nonzero sum Heffter arrays and biembeddings

Square relative nonzero sum Heffter arrays, denoted by N H t ( n ; k ) ${\rm{N}}{{\rm{H}}}_{t}(n;k)$, have been introduced as a variant of the classical concept of Heffter array. An N H t ( n ; k ) ${\rm{N}}{{\rm{H}}}_{t}(n;k)$ is an n x n $n\times n$ partially filled array with elements in Z v ${{\mathbb{Z}}}_{v}$, where v = 2 n k + t $v=2nk+t$, whose rows and whose columns contain k $k$ filled cells, such that the sum of the elements in every row and column is different from 0 (modulo v $v$) and, for every x is an element of Z v $x\in {{\mathbb{Z}}}_{v}$ not belonging to the subgroup of order t $t$, either x $x$ or - x $-x$ appears in the array. In this paper we give direct constructions of square nonzero sum Heffter arrays with no empty cells, N H t ( n ; n ) ${\rm{N}}{{\rm{H}}}_{t}(n;n)$, for every n $n$ odd, when t $t$ is a divisor of n $n$ and when t is an element of { 2 , 2 n , n 2 , 2 n 2 } $t\in \{2,2n,{n}<^>{2},2{n}<^>{2}\}$. 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.