Tight globally simple non-zero sum Heffter arrays and biembeddings

TL;DR

This paper introduces new constructions of square non-zero sum Heffter arrays that are globally simple, enabling the creation of novel orthogonal path decompositions and biembeddings of complete multipartite graphs.

Contribution

It provides explicit constructions of globally simple non-zero sum Heffter arrays for specific parameters, expanding their applications in graph embeddings and decompositions.

Findings

Constructed arrays for all odd n with t dividing n

Arrays have applications in orthogonal path decompositions

Enabled new biembeddings of complete multipartite graphs

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.