Sequenceable Groups and Related Topics

Ollis, M. A.

doi:10.37236/30

Cited by 26 publications

(21 citation statements)

References 76 publications

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“…Clearly, when k = |G| − 3, |G| − 2, |G| − 1, G is assumed to be finite. Alspach's conjecture is worth to be studied also in connection with sequenceability and strong sequenceability of groups, see [2,3,18], and simplicity of Heffter arrays, see [4,5,12,14].…”

Section: Introductionmentioning

confidence: 99%

Some new results about a conjecture by Brian Alspach

Costa

Pellegrini

2020

Arch. Math.

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In this paper, we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset A of $$\mathbb {Z}_n{\setminus } \{0\}$$ Z n \ { 0 } of size k such that $$\sum _{z\in A} z\not = 0$$ ∑ z ∈ A z ≠ 0 , it is possible to find an ordering $$(a_1,\ldots ,a_k)$$ ( a 1 , … , a k ) of the elements of A such that the partial sums $$s_i=\sum _{j=1}^i a_j$$ s i = ∑ j = 1 i a j , $$i=1,\ldots ,k$$ i = 1 , … , k , are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $$k\le 11$$ k ≤ 11 in cyclic groups of prime order. Here, we extend this result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $$\mathbb {Z}_n$$ Z n . We also consider a related conjecture, originally proposed by Ronald Graham: given a subset A of $$\mathbb {Z}_p{\setminus }\{0\}$$ Z p \ { 0 } , where p is a prime, there exists an ordering of the elements of A such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis, and Schmitt, based on Alon’s combinatorial Nullstellensatz, we prove the validity of this conjecture for subsets A of size 12.

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Section: Introductionmentioning

confidence: 99%

Some new results about a conjecture by Brian Alspach

Costa

Pellegrini

2020

Arch. Math.

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“…As far as we know, Conjecture 7 has not been attacked directly. However, many partial results are known due to its connection to sequenceable groups (see eg, [20]).…”

Section: Discussionmentioning

confidence: 99%

All group‐based latin squares possess near transversals

Goddyn

Halasz

2020

J of Combinatorial Designs

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“…In addition to those already mentioned, groups known to satisfy Keedwell's Conjecture include dihedral groups [15,18], soluble groups with a single involution [3], and groups of order at most 255 [22]. See [20] for a survey of this and related problems.…”

Section: Introductionmentioning

confidence: 99%