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Relative Heffter arrays and biembeddings

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 $n$.