A generalization of Heffter arrays

In 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 v = 2nk + t be a positive integer, where t divides 2nk, and let J be the subgroup of Zv of order t. A Ht (m, n; s, k) Heffter array over Zv relative to J is an m × n partially filled array with elements in Zv such that (a) each row contains S filled cells and each column contains k filled cells; (b) for every x ∈ ZvJ, either x or -x appears in the array; and (c) the elements in every row and column sum to 0. Here we study the existence of square integer (i.e., with entries chosen in (Formula presented.) and where the sums are zero in Z) relative Heffter arrays for t = k, denoted by Hk (n; k).. In particular, we prove that for 3 ≤ k ≤ n, with k ≠ 5, there exists an integer Hk (n; k) if and only if one of the following holds: (a) k is odd and n ≡ 0, 3 (mod 4); (b) k ≡ 2 (mod 4) and n is even; (c) k ≡ 0 (mod 4). Also, we show how these arrays give rise to cyclic cycle decompositions of the complete multipartite graph.