TY - JOUR

T1 - A generalization of Heffter arrays

AU - Pellegrini, Marco Antonio

AU - Costa, Simone

AU - Morini, Fiorenza

AU - Pasotti, Anita

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Heffter array

KW - multipartite complete graph

KW - orthogonal cyclic cycle decomposition

KW - Heffter array

KW - multipartite complete graph

KW - orthogonal cyclic cycle decomposition

UR - http://hdl.handle.net/10807/150156

U2 - 10.1002/jcd.21684

DO - 10.1002/jcd.21684

M3 - Article

VL - 28

SP - 171

EP - 206

JO - Journal of Combinatorial Designs

JF - Journal of Combinatorial Designs

SN - 1063-8539

ER -