A survey of Heffter arrays

Pasotti, A.; Dinitz, J. H.

doi:10.48550/arxiv.2209.13879

Cited by 3 publications

(6 citation statements)

References 36 publications

(105 reference statements)

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“…Also, the factors of an optimal rank-one H (3,8) over F 49 are the following: X = {1, 2, 4}; Y = {1, 2g, 1 + 5g, 5 + 6g, 5 + 3g, 6 + g, 4 + 5g, 6 + 6g}. Now let g be a root of the primitive polynomial z 4 + z + 2 over F 3 and, to save space, let us write each element a + bg + cg 2 + dg 3 of F 81 in the form abcd.…”

Section: Rank-one Heffter Arrays With (N K) Agreeablementioning

confidence: 99%

“…Now let g be a root of the primitive polynomial z 4 + z + 2 over F 3 and, to save space, let us write each element a + bg + cg 2 + dg 3 of F 81 in the form abcd. Then the factors of an optimal rank-one H (5,8) Let us investigate if there are conditions assuring the existence of the first ingredient X requested by Lemma 6.1. Lemma 6.3.…”

Section: Rank-one Heffter Arrays With (N K) Agreeablementioning

confidence: 99%

“…Heffter arrays are interesting combinatorial designs introduced by Archdeacon [1] in 2015. For a rich survey on the many variations and the related results we refer to [8]. In this note we consider Heffter arrays in the original meaning of "tight" Heffter arrays extending the definition to any group of odd order as proposed in [7].…”

Section: Introductionmentioning

confidence: 99%

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Tight Heffter arrays from finite fields

Buratti¹

2022

Preprint

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After extending in the obvious way the classic notion of a Heffter array H(n, k) to any group of order 2nk + 1, we give direct constructions for elementary abelian Heffter arrays, hence in particular for prime Heffter arrays (whose existence was already known). If q = 2nk + 1 is a prime power, we say that an elementary abelian H(n, k) is "over Fq" since, for its construction, we exploit both the additive and multiplicative structure of the field of order q.We show that in many cases a direct construction of a H(n, k) over Fq, say A, can be obtained very easily by imposing that A has rank 1 and, possibly, a rich group of multipliers, that are elements m of Fq such that mA = A up to a permutation of rows and columns. A H(n, k) over Fq will be said optimal if the order of its group of multipliers is the least common multiple of the odd parts of n and k, since this is the maximum possible order for it.We give an explicit direct construction of a rank-one H(n, k) -reaching almost always the optimality -for all admissible pairs (n, k) except for the demanding ones, i.e., those where n or k is a power of 2, or nk has only one odd prime divisor. A less explicit construction makes reasonable to believe that an optimal rank-one H(n, k) exists for any admissible pair (n, k).

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Section: Rank-one Heffter Arrays With (N K) Agreeablementioning

confidence: 99%

Section: Rank-one Heffter Arrays With (N K) Agreeablementioning

confidence: 99%

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Tight Heffter arrays from finite fields

Buratti¹

2022

Preprint

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“…Combining several results and conjectures concering subsets of abelian groups dating back to 1961 [8], we have the conjecture that all abelian groups are strongly sequenceable. See [2,5,18] for details of the history, motivation and current state of this conjecture.…”

Section: Introductionmentioning

confidence: 99%

Sequencings in Semidirect Products via the Polynomial Method

Costa¹,

Fiore²,

Ollis³

2023

Preprint

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The partial sums of a sequence x = x 1 , x 2 , . . . , x k of distinct non-identity elements of a group (G, •) are s 0 = id G and s j = j i=1 x i for 0 < j ≤ k. If the partial sums are all different then x is a linear sequencing and if the partial sums are all different when |i−j| ≤ t then x is a t-weak sequencing. We investigate these notions of sequenceability in semidirect products using the polynomial method. We show that every subset of order k of the non-identity elements of the dihedral group of order 2m has a linear sequencing when k ≤ 12 and either m > 3 is prime or every prime factor of m is larger than k!, unless s k is unavoidably the identity; that every subset of order k of a non-abelian group of order three times a prime has a linear sequencing when 5 < k ≤ 10, unless s k is unavoidably the identity; and that if the order of a group is pe then all sufficiently large subsets of the non-identity elements are t-weakly sequenceable when p > 3 is prime, e ≤ 3 and t ≤ 6.

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“…If the boundary of a face is homeomorphic to a circumference (i.e. is a simple cycle), such a face is said to be simple and if all the faces are simple we say that the embedding is circular (or simple, following the notation of [32]). If moreover, the embedding is face-2-colourable, sometimes with abuse of notation, we call it a biembedding.…”

Section: Introductionmentioning

confidence: 99%

On the number of non-isomorphic (simple) k-gonal biembeddings of complete multipartite graphs

Costa,

Pasotti

2023

Ars Math. Contemp.

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This article aims to provide exponential lower bounds on the number of non-isomorphic k-gonal face-2-colourable embeddings (sometimes called, with abuse of notation, biembeddings) of the complete multipartite graph into orientable surfaces.For this purpose, we use the concept, introduced by Archdeacon in 2015, of Heffer array and its relations with graph embeddings. In particular we show that, under certain hypotheses, from a single Heffter array, we can obtain an exponential number of distinct graph embeddings. Exploiting this idea starting from the arrays constructed by Cavenagh, Donovan and Yazıcı in 2020, we obtain that, for infinitely many values of k and v, therewhere H(•) is the binary entropy. Moreover about the embeddings of K v t ×t , for t ∈ {1, 2, k}, we provide a construction of 2 v• H(1/4) 2k(k−1) +o(v,k) non-isomorphic k-gonal face-2-colourable embeddings whenever k is odd and v belongs to a wide infinite family of values.

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A survey of Heffter arrays

Abstract: Dedicated to Doug Stinson on the occasion of his 66th birthday.

Cited by 3 publications

References 36 publications

Tight Heffter arrays from finite fields

Tight Heffter arrays from finite fields

Sequencings in Semidirect Products via the Polynomial Method

On the number of non-isomorphic (simple) k-gonal biembeddings of complete multipartite graphs

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