Langford sequences: perfect and hooked

Simpson, James

doi:10.1016/0012-365x(83)90008-0

Cited by 93 publications

(102 citation statements)

References 6 publications

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“…He showed that for every length l either a Langford or a hooked Langford sequence of defect d = 2 exists and he conjectured the necessary conditions for their existence. The generalization of the problem for any defect d was partially solved by D a v i e s [26] and B e r m o n d et al [12], and it was completed by S i m p s o n [96].…”

Section: ì óö ñ 214º ([91]) a K-near Skolem Sequence Of Order N Exismentioning

confidence: 99%

A survey of Skolem-type sequences and Rosa’s use of them

Francetić

Mendelsohn

2009

Mathematica Slovaca

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Dedicated to Alex Rosa on the occasion of his seventieth birthday May he live in full strength (as Moses did) until one hundred twenty (Communicated by Peter Horák )ABSTRACT. Let D be a set of positive integers. A Skolem-type sequence is a sequence of i ∈ D such that every i ∈ D appears exactly twice in the sequence at positions a i and b i , and |b i − a i | = i. These sequences might contain empty positions, which are filled with null elements. Thoralf A. Skolem defined and studied Skolem sequences in order to generate solutions to Heffter's difference problems. Later, Skolem sequences were generalized in many ways to suit constructions of different combinatorial designs. Alexander Rosa made the use of these generalizations into a fine art. Here we give a survey of Skolem-type sequences and their applications.

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Section: ì óö ñ 214º ([91]) a K-near Skolem Sequence Of Order N Exismentioning

confidence: 99%

A survey of Skolem-type sequences and Rosa’s use of them

Francetić

Mendelsohn

2009

Mathematica Slovaca

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“…For t ≡ 0, 1 (mod 4) (respectively t ≡ 2, 3 (mod 4)), we can obtain a (3 tn )-decomposition of S t n by using a Langford sequence (respectively hooked Langford sequence) of order t and defect 5, which exists since t ≥ 9, to partition S t into difference triples (see [90,91]). So we have a (3 tn )-decomposition of S t n , and require a (4 qn , n h )-decomposition of K n − {1, 2, 3, 4, 6}∪S t n .…”

Section: T=1mentioning

confidence: 99%

“…We then add 1 to each element of each of these triples to obtain a partition of {5, . [90,91]), and use it to partition {4, . .…”

Section: Proof Of Lemma 137mentioning

confidence: 99%

Computational Graph Theory

Tinhofer¹,

Mayr²,

Noltemeier³

et al. 1990

Computing Supplementum

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This thesis involves the application of computational techniques to various problems in graph theory and low dimensional topology. The first two chapters of this thesis focus on problems in graph theory itself; in particular on graph decomposition problems. The last three chapters look at applications of graph theory to combinatorial topology, focusing on the exhaustive generation of certain families of 3-manifold triangulations.

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“…In 1983, Simpson proved sufficient conditions for existence of perfect Langford sequences with the addition of the defect d variable indicating the lower bound of the range of integers to include in the sequence [22]. He also showed necessary and sufficient conditions for the hooked versions of these Langford sequences [22].…”

Section: Introductionmentioning

confidence: 99%

“…He also showed necessary and sufficient conditions for the hooked versions of these Langford sequences [22].…”

Section: Introductionmentioning

confidence: 99%

Generalized Langford sequences: new results and algorithms

Mata-Montero¹,

Normore²,

Shalaby³

2014

imf

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This paper presents new characterizations and algorithmic results for generalized Skolem and Langford sequences with minimum number of zeros. It generalize the necessary conditions for existence and locations for each zero of generalized Langford sequences. New algorithm implementations are used to find new sequences and analyze the space of solutions for a range of cases. Expanded tables of computational results are given, along with several efficient probabilistic ways of finding solutions in very large search cases.Mathematics Subject Classification: 05B

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Langford sequences: perfect and hooked

Cited by 93 publications

References 6 publications

A survey of Skolem-type sequences and Rosa’s use of them

A survey of Skolem-type sequences and Rosa’s use of them

Computational Graph Theory

Generalized Langford sequences: new results and algorithms

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