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 = 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 ∈ 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_{(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.