Rectangular Heffter arrays: a reduction theorem

Morini, Fiorenza; Pellegrini, Marco Antonio

doi:10.1016/j.disc.2022.113073

Cited by 6 publications

(17 citation statements)

References 15 publications

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“…Existence results on signed magic arrays can be found in [53,54,55,56,57,64,65]. We note that nearly all the results and techniques used on signed magic arrays are analogous to those for Heffter arrays.…”

Section: Signed Magic Arraysmentioning

confidence: 80%

“…Recent partial results for the general case can be found in [65]. We summarize them in the following two theorems.…”

Section: Signed Magic Arraysmentioning

confidence: 91%

“…We note here that the necesary conditions for the existence of an integer H(m, n; h, k) are that 3 ≤ h ≤ n, 3 ≤ k ≤ m, mh = nk and nk ≡ 0, 3 (mod 4). Recent partial results for integer rectangular arrays with empty cells can be found in [65]. We summarize these results in the following two theorems.…”

Section: Introductionmentioning

confidence: 90%

“…Theorem 6.24 [35] If there exists an α H t (m, n; h, k) then, for any divisor λ of t, there exists a λα H t λ (m, n; h, k). Theorem 6.25 [65] Let m, n, h, k be such that 2 ≤ h ≤ n, 2 ≤ k ≤ m and mh = nk. Let λ be a divisor of 2nk and let t be a divisor of 2nk λ .…”

Section: Integer Relative Heffter Arrays With λ =mentioning

confidence: 99%

“…Theorem 1.18 [65] Let m, n, h, k be such that 3 ≤ h ≤ n, 3 ≤ k ≤ m and mh = nk. If h ≡ 0 (mod 4) and k = 5 is odd, then there exists an integer H(m, n; h, k).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A survey of Heffter arrays

Pasotti¹,

Dinitz²

2022

Preprint

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Section: Signed Magic Arraysmentioning

confidence: 80%

“…Recent partial results for the general case can be found in [65]. We summarize them in the following two theorems.…”

Section: Signed Magic Arraysmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 90%

Section: Integer Relative Heffter Arrays With λ =mentioning

confidence: 99%

“…Theorem 1.18 [65] Let m, n, h, k be such that 3 ≤ h ≤ n, 3 ≤ k ≤ m and mh = nk. If h ≡ 0 (mod 4) and k = 5 is odd, then there exists an integer H(m, n; h, k).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A survey of Heffter arrays

Pasotti¹,

Dinitz²

2022

Preprint

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Magic partially filled arrays on abelian groups

Morini

Pellegrini

2023

J of Combinatorial Designs

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In this paper we introduce a special class of partially filled arrays. A magic partially filled array MPFΩ(m,n;s,k) ${\text{MPF}}_{{\rm{\Omega }}}(m,n;s,k)$ on a subset normalΩ ${\rm{\Omega }}$ of an abelian group (normalΓ,+) $({\rm{\Gamma }},+)$ is a partially filled array of size m×n $m\times n$ with entries in normalΩ ${\rm{\Omega }}$ such that (i) every ω∈normalΩ $\omega \in {\rm{\Omega }}$ appears once in the array; (ii) each row contains s $s$ filled cells and each column contains k $k$ filled cells; (iii) there exist (not necessarily distinct) elements x,y∈normalΓ $x,y\in {\rm{\Gamma }}$ such that the sum of the elements in each row is x $x$ and the sum of the elements in each column is y $y$. In particular, if x=y=0Γ $x=y={0}_{{\rm{\Gamma }}}$, we have a zero‐sum magic partially filled array MPFΩ0(m,n;s,k) ${}^{0}\text{MPF}_{{\rm{\Omega }}}(m,n;s,k)$. Examples of these objects are magic rectangles, normalΓ ${\rm{\Gamma }}$‐magic rectangles, signed magic arrays, (integer or noninteger) Heffter arrays. Here, we give necessary and sufficient conditions for the existence of a magic rectangle with empty cells, that is, of an MPFΩ(m,n;s,k) ${\text{MPF}}_{{\rm{\Omega }}}(m,n;s,k)$ where normalΩ={1,2,…,nk}⊂normalℤ ${\rm{\Omega }}=\{1,2,\ldots ,nk\}\subset {\rm{{\mathbb{Z}}}}$. We also construct zero‐sum magic partially filled arrays when normalΩ ${\rm{\Omega }}$ is the abelian group normalΓ ${\rm{\Gamma }}$ or the set of its nonzero elements.

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Orthogonal cycle systems with cycle length less than 10

Küçükçifçi,

Yazıcı

2023

J of Combinatorial Designs

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An ‐decomposition of a graph is a partition of the edge set of into subsets, where each subset induces a copy of the graph . A ‐orthogonal ‐decomposition of is a set of ‐decompositions of such that any two copies of in distinct ‐decompositions intersect in at most one edge. When , we call the ‐decomposition an ‐system of order . In this paper, we consider the case is an ‐cycle and construct a pair of orthogonal ‐cycle systems for all admissible orders when , except when .

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Rectangular Heffter arrays: a reduction theorem

Cited by 6 publications

References 15 publications

A survey of Heffter arrays

A survey of Heffter arrays

Magic partially filled arrays on abelian groups

Orthogonal cycle systems with cycle length less than 10

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