Relative Heffter arrays and biembeddings

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 x n partially filled array with elements in 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 is an element of 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 K2nk+t /t x 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.