Geodesic
Relative Heffter arrays and biembeddings
Ars Mathematica Contemporanea, Tome 18 (2020) no. 2, pp. 241-271.

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Relative Heffter arrays, denoted by Ht(m, n; s, k), have been introduced as a generalization of the classical concept of Heffter array. A Ht(m, n; s, k) is an m × n partially filled array with elements in ℤv, 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 ∈ ℤv 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(2nk + t)/t × 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 ≡ 3 (mod 4) and for k = 3 with t = n, 2n, any odd n.
DOI : 10.26493/1855-3974.2110.6f2
Keywords: Heffter array, biembedding, multipartite complete graph 
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Simone Costa; Anita Pasotti; Marco Antonio Pellegrini. Relative Heffter arrays and biembeddings. Ars Mathematica Contemporanea, Tome 18 (2020) no. 2, pp. 241-271. doi : 10.26493/1855-3974.2110.6f2. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2110.6f2/

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