Relative Heffter arrays and biembeddings

TL;DR

This paper introduces relative Heffter arrays as a generalization of classical Heffter arrays and demonstrates their use in constructing biembeddings of cyclic cycle decompositions of complete multipartite graphs into orientable surfaces.

Contribution

The paper develops new constructions of biembeddings using relative Heffter arrays, expanding the applications of these arrays in topological graph theory.

Findings

Constructed biembeddings for specific parameters $t=k=3,5,7,9$ and $n\equiv 3 \pmod 4$.

Provided integer globally simple square relative Heffter arrays for certain parameters.

Extended the use of relative Heffter arrays to embed cyclic decompositions into surfaces.

Abstract

Relative Heffter arrays, denoted by $\mathrm{H}_t(m,n; s,k)$, have been introduced as a generalization of the classical concept of Heffter array. A $\mathrm{H}_t(m,n; s,k)$ is an $m\times n$ partially filled array with elements in $\mathbb{Z}_v$, where $v=2nk+t$, whose rows contain $s$ filled cells and whose columns contain $k$ filled cells, such that the elements in every row and column sum to zero 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 show how relative Heffter arrays can be used to construct biembeddings of cyclic cycle decompositions of the complete multipartite graph $K_{\frac{2nk+t}{t}\times t}$ into an orientable surface. In particular, we construct such biembeddings providing integer globally simple square relative Heffter arrays for $t=k=3,5,7,9$ and $n\equiv 3 \pmod 4$ and for $k=3$ with $t=n,2n$, any odd $n$.