A generalization of Heffter arrays

Abstract

AbstractIn this paper, we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let be a positive integer, where divides , and let be the subgroup of of order . A Heffter array over relative to is an partially filled array with elements in such that (a) each row contains filled cells and each column contains filled cells; (b) for every , either or appears in the array; and (c) the elements in every row and column sum to . Here we study the existence of square integer (i.e., with entries chosen in and where the sums are zero in ) relative Heffter arrays for , denoted by . In particular, we prove that for , with , there exists an integer if and only if one of the following holds: (a) is odd and ; (b) and is even; (c) . Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.