On the existence of integer relative Heffter arrays

Morini, Fiorenza; Pellegrini, Marco Antonio

doi:10.1016/j.disc.2020.112088

Cited by 13 publications

(15 citation statements)

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“…The first aim of this paper is to show how it is possible to reduce the problem of the existence of a rectangular Heffter array to the square case where the elements belong to consecutive diagonals (see Theorem 3.3). This process, combined with [22,Theorem 1.3], allows us to prove the following.…”

Section: Introductionmentioning

confidence: 89%

“…Proof. We recall that there exists an integer diagonal H(a; b), with a ≥ b ≥ 3, in each of the following cases: (i) b ≡ 0 (mod 4) (shiftable), see [3] and [22];…”

Section: The Reduction Theoremmentioning

confidence: 99%

“…One of the main reasons that justifies the study of Heffter arrays and of their generalizations is that they allow, under suitable conditions (or assuming the validity of [8, Conjecture 3]), to produce biembeddings of pairs of cyclic decompositions (one decomposition consisting of s-cycles and the other one consisting of k-cycles) of the complete multipartite multigraph λ K ℓ×t , with ℓ parts each of size t, onto orientable surfaces (see [5,7,9,11,12,13]). Partial results about the existence of these generalizations have been obtained in [10,11,12,22,23].…”

Section: Introductionmentioning

confidence: 99%

“…To obtain our block Z i we use the block U i for the first three rows, and we need a shiftable H(k − 3, 4; 4, k − 3) to construct the remaining k − 3 rows. The existence of this Heffter array was proved in [2,22], but here, for the reader's sake, we prefer to give an explicit construction. So, consider the following shiftable blocks: Note that Q 4 is an integer H(4, 4; 4, 4) and Q 6 is an integer H (6, 4; 4, 6).…”

mentioning

confidence: 99%

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Rectangular Heffter arrays: a reduction theorem

Morini¹,

Pellegrini²

2021

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Let m, n, s, k be four integers such that 3 ≤ s ≤ n, 3 ≤ k ≤ m and ms = nk. Set d = gcd(s, k). In this paper we show how one can construct a Heffter array H(m, n; s, k) starting from a square Heffter array H(nk/d; d) whose elements belong to d consecutive diagonals. As an example of application of this method, we prove that there exists an integer H(m, n; s, k) in each of the following cases: (i) d ≡ 0 (mod 4); (ii) 5 ≤ d ≡ 1 (mod 4) and nk ≡ 3 (mod 4); (iii) d ≡ 2 (mod 4) and nk ≡ 0 (mod 4); (iv) d ≡ 3 (mod 4) and nk ≡ 0, 3 (mod 4). The same method can be applied also for signed magic arrays SMA(m, n; s, k) and for magic rectangles MR(m, n; s, k). In fact, we prove that there exists an SMA(m, n; s, k) when d ≥ 2, and there exists an MR(m, n; s, k) when either d ≥ 2 is even or d ≥ 3 and nk are odd. We also provide constructions of integer Heffter arrays and signed magic arrays when k is odd and s ≡ 0 (mod 4).

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Section: Introductionmentioning

confidence: 89%

“…Proof. We recall that there exists an integer diagonal H(a; b), with a ≥ b ≥ 3, in each of the following cases: (i) b ≡ 0 (mod 4) (shiftable), see [3] and [22];…”

Section: The Reduction Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Rectangular Heffter arrays: a reduction theorem

Morini¹,

Pellegrini²

2021

Preprint

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“…In particular, in [10] it was proved that Heffter arrays H 1 (n; k) exist for all n ≥ k ≥ 3, while by [4,17] integer Heffter arrays H 1 (n; k) exist if and only if the additional condition nk ≡ 0, 3 (mod 4) holds. At the moment, the only known results concerning relative Heffter arrays are described in [15,22]. Some necessary conditions for the existence of an integer H t (n; k) are given by the following.…”

Section: Introductionmentioning

confidence: 99%

Relative Heffter arrays and biembeddings

Costa¹,

Pasotti²,

Pellegrini³

2019

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Relative Heffter arrays, denoted by Ht(m, n; s, k), have been introduced as a generalization of the classical concept of Heffter array. A Ht(m, n; s, k) is an m × n partially filled array with elements in Zv, where v = 2nk + t, whose rows contain s filled cells and whose columns contain k filled cells, such that the elements in every row and column sum to zero and, for every x ∈ Zv not belonging to the subgroup of order t, either x or −x appears in the array. In this paper we show how relative Heffter arrays can be used to construct biembeddings of cyclic cycle decompositions of the complete multipartite graph K 2nk+t t ×t into an orientable surface. In particular, we construct such biembeddings providing integer globally simple square relative Heffter arrays for t = k = 3, 5, 7, 9 and n ≡ 3 (mod 4) and for k = 3 with t = n, 2n, any odd n.

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