Hooked k-extended Skolem sequences

Linek, Václav; Jiang, Zhiying

doi:10.1016/s0012-365x(98)00202-7

Cited by 13 publications

(23 citation statements)

References 7 publications

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“…The sufficiency of these conditions is proved by exhibiting direct constructions of these systems using Skolem-type sequences (cf. [50]), similarly as in the above proofs. In particular, L i n e k and J i a n g applied most of the already defined Skolem related sequences.…”

Section: ì óö ñ 36º ([70]) There Exists a 2-rotational St S(v) If Ansupporting

confidence: 55%

“…[6]). The k-extended and hooked k-extended Langford sequences were studied by L i n e k and J i a n g [49], [50]. Similar to the definitions above, a k-extended Langford sequence of defect d and length l is a k-extended Skolem-type sequence with m = 2l+1 and…”

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

confidence: 99%

“…Moreover, L i n e k and J i a n g [50] present several direct constructions of hooked extended Skolem sequences by taking a sum (union) of two Skolemtype sequences. Let null elements in the sequence be represented by 0.…”

Section: Techniques Used In the Proofs Of The Existence Theoremsmentioning

confidence: 99%

“…Also, substitute the second and the third occurrence of i in S with the appropriate difference. Hence, the resulting sequence is a Skolem-type sequence with differences 2l The hooked k-extended Skolem sequences and their existence were studied by L i n e k and J i a n g [50]. They obtained the required sequences of small orders by a computer program based on the hill-climbing algorithm ( [95]).…”

Section: Techniques Used In the Proofs Of The Existence Theoremsmentioning

confidence: 99%

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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 36º ([70]) There Exists a 2-rotational St S(v) If Ansupporting

confidence: 55%

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

confidence: 99%

Section: Techniques Used In the Proofs Of The Existence Theoremsmentioning

confidence: 99%

Section: Techniques Used In the Proofs Of The Existence Theoremsmentioning

confidence: 99%

See 2 more Smart Citations

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

Francetić

Mendelsohn

2009

Mathematica Slovaca

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“…(4, 2)) and k ≥ m + 2. Also, all k-extensions can be obtained for hooked Langford sequences with 1 ≤ d ≤ 2 and differences in D j , 6 ≤ j ≤ 8, and this is easily extended by hand calculation to d = 3 (see Linek and Jiang [4], [5]) Henceforth, we take d ≥ 4.…”

Section: The Results For Differencesmentioning

confidence: 99%

Hooked extended Langford sequences of small and large defects

Mor

Linek

2014

Mathematica Slovaca

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ABSTRACT. It is shown that for m = 2d + 5, 2d + 6, 2d + 7 and d ≥ 1, the set {1, . . . , 2m + 1} − {k} can be partitioned into differencesIt is also shown that for m = 2d + 5, 2d + 6, 2d + 7, and d ≥ 1, the set {1, . . . , 2m + 2} − {k, 2m + 1} can be partitioned into differencesThese partitions are used to show that if m ≥ 8d + 3, then the set {1, . . .A list of values m, d that are open for the existence of these partitions (which are equivalent to the existence of Langford and hooked Langford sequences) is given in the conclusion. The juxtaposition, S 1 S 2 , of sequences S 1 and S 2 is their concatenation. IntroductionFor general information about Langford sequences we refer to Shalaby [12].

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On partitions of {1,…,2m+ 1}\{k} into differencesd,…,d+m− 1: Extended Langford sequences of large defect

Linek

Mor

2004

J of Combinatorial Designs

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It is shown that for m = 2d − 1, 2d, 2d + 1, and d ≥ 1, the set {1, 2,…, 2m + 2}, − {2,k} can be partitioned into differences d,d + 1,…,d + m − 1 whenever (m,k) ≡ (0,0), (1,d + 1), (2, 1), (3,d) (mod (4,2)) and (d,m,k) ≠ (1,1,3), (2,3,7) (where (x,y) ≡ (u,ν) mod (m,n) iff x ≡ u (mod m) and y ≡ ν (mod n)). It is also shown that if m ≥ 2d − 1 and m ∉ [2d + 2, 8d − 5], then the set {1, 2, …, 2m + 1} − {k} can be partitioned into differences d,d + 1,…,d + m − 1 whenever (m,k) ≡ (0, 1), (1,d), (2,0), (3,d + 1) mod (4,2). Finally, for d = 4 we obtain a complete result for when {1,…,2m + 1} − {k} can be partitioned into differences 4,5,…,m + 3. © 2004 Wiley Periodicals, Inc.

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Hooked k-extended Skolem sequences

Cited by 13 publications

References 7 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

Hooked extended Langford sequences of small and large defects

On partitions of {1,…,2m+ 1}\{k} into differencesd,…,d+m− 1: Extended Langford sequences of large defect

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