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.

Costa, S., Morini, F., Pasotti, A., Pellegrini, M. A., A generalization of Heffter arrays, <<JOURNAL OF COMBINATORIAL DESIGNS>>, 2020; 28 (3): 171-206. [doi:10.1002/jcd.21684] [http://hdl.handle.net/10807/150156]

A generalization of Heffter arrays

Pellegrini, M. A.
Ultimo
2020

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 = 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.
Inglese
Costa, S., Morini, F., Pasotti, A., Pellegrini, M. A., A generalization of Heffter arrays, <<JOURNAL OF COMBINATORIAL DESIGNS>>, 2020; 28 (3): 171-206. [doi:10.1002/jcd.21684] [http://hdl.handle.net/10807/150156]
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/150156
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 10
  • ???jsp.display-item.citation.isi??? 10
social impact