A generalization of Heffter arrays. Issue 3 (19th November 2019) - Costa, Simone; Morini, Fiorenza; Pasotti, Anita; Pellegrini, Marco Antonio
Abstract: 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 = 2 n k + t be a positive integer, where t divides 2 n k, and let J be the subgroup of Z v of order t . A H t ( m, n ; s, k ) Heffter array over Z v relative to J is an m × n partially filled array with elements in Z v such that (a) each row contains s filled cells and each column contains k filled cells; (b) for every x ∈ Z v \ J, 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 ± 1, …, 2 n k + t 2 and where the sums are zero in Z ) relative Heffter arrays for t = k, denoted by H k ( n ; k ) . In particular, we prove that for 3 ≤ k ≤ n, with k ≠ 5, there exists an integer H k ( 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.