Tight globally simple nonzero sum Heffter arrays and biembeddings

Abstract

AbstractSquare relative nonzero sum Heffter arrays, denoted by , have been introduced as a variant of the classical concept of Heffter array. An is an partially filled array with elements in , where , whose rows and whose columns contain filled cells, such that the sum of the elements in every row and column is different from 0 (modulo ) and, for every not belonging to the subgroup of order , either or appears in the array. In this paper we give direct constructions of square nonzero sum Heffter arrays with no empty cells, , for every odd, when is a divisor of and when . The constructed arrays have also the very restrictive property of being “globally simple”; this allows us to get new orthogonal path decompositions and new biembeddings of complete multipartite graphs.