Non-zero sum Heffter arrays and their applications

TL;DR

This paper introduces non-zero sum Heffter arrays, explores their mathematical properties and connections to graph theory, and demonstrates their applications in constructing biembeddings of complete graphs.

Contribution

It defines a new class of arrays related to difference families and graph decompositions, providing existence results and constructions, especially for square and complete rectangular cases.

Findings

Complete existence solutions for non-zero sum Heffter arrays.

Constructive methods for square and rectangular arrays with no empty cells.

Applications in biembedding complete graphs.

Abstract

In this paper we introduce a new class of partially filled arrays that, as Heffter arrays, are related to difference families, graph decompositions and biembeddings. A non-zero sum Heffter array $\mathrm{N}\mathrm{H}(m,n; h,k)$ is an $m \times n$ p. f. array with entries in $\mathbb{Z}_{2nk+1}$ such that: each row contains $h$ filled cells and each column contains $k$ filled cells; for every $x\in \mathbb{Z}_{2nk+1}\setminus\{0\}$, either $x$ or $-x$ appears in the array; the sum of the elements in every row and column is different from $0$ (in $\mathbb{Z}_{2nk+1}$). Here first we explain the connections with relative difference families and with path decompositions of the complete multipartite graph. Then we present a complete solution for the existence problem and a constructive complete solution for the square case and for the rectangular case with no empty cells when the additional, very restrictive, property of "globally simple" is required. Finally, we show how these arrays can be used to construct biembeddings of complete graphs.