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 (2 n k + 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 , 2 n , any odd n .