### Relative Heffter arrays and biembeddings

#### Abstract

Relative Heffter arrays, denoted by *H*_{t}(*m*, *n*; *s*, *k*), have been introduced as a generalization of the classical concept of Heffter array. A *H*_{t}(*m*, *n*; *s*, *k*) is an *m* × *n* partially filled array with elements in ℤ_{v}, where *v* = 2*n**k* + *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* ∈ 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*, 2*n*, any odd *n*.

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MANUSCRIPTISSN: 1855-3974

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