The existence of near-Skolem and hooked near-Skolem sequences

Shalaby, Nabil

doi:10.1016/0012-365x(92)00327-n

Cited by 25 publications

(31 citation statements)

References 7 publications

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“…Clearly, q = 6. A hooked Skolem sequence of order 6 (in ordered pair notation) is given by S = {(10, 11), (2,4), (6,9), (1,5), (3,8), (7,13), (0, 12)}. The corresponding antimagic labeling of the digraph G * is given in Figure 2.…”

Section: Resultsmentioning

confidence: 99%

Antimagic Labeling of Digraphs

Moviri

2016

JIMS

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Abstract. An antimagic labeling of a digraph D with p vertices and q arcs is a bijection f from the set of all arcs to the set of positive integers {1, 2, 3, ..., q} such that all the p oriented vertex weights are distinct, where an oriented vertex weight is the sum of the labels of all arcs entering that vertex minus the sum of the labels of all arcs leaving it. A digraph D is called antimagic if it admits an antimagic labeling. In this paper we investigate the existence of antimagic labelings of some few families of digraphs using hooked Skolem sequences.Key words and Phrases: Antimagic labeling, hooked Skolem sequence, symmetric digraph.Abstrak. Suatu pelabelan antimagic dari sebuah digraf D dengan p buah titik dan q buah busur adalah suatu bijeksi f dari himpunan semua busur ke himpunan bilangan bulat positif {1, 2, 3, ..., q} sedemikian sehingga semua bobot titik berarah p berbeda, dimana suatu bobot titik berarah adalah jumlahan label-label dari semua busur yang menuju titik tersebut dikurangi jumlahan label-label dari semua busur yang keluar dari titik tersebut. Sebuah digraf D dikatakan antimagic jika mempunyai pelabelan antimagic. Dalam paper ini kami menunjukkan keberadaan pelabelan antimagic dari beberapa keluarga digraf menggunakan barisan Skolem terkait.Kata kunci: Pelabelan antimagic, barisan Skolem terkait, digraf simetris.

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Section: Resultsmentioning

confidence: 99%

Antimagic Labeling of Digraphs

Moviri

2016

JIMS

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“…The necessary conditions for the existence of these sequences are derived in [91] and their sufficiency is proved in [6]. In particular, if k and n are such that the conditions of the existence of an excess Skolem sequence hold, then there exists a k-extended Skolem sequence of order n, and we can add a surplus copy of k's as the first element of the sequence and in the place of the hook in the extended Skolem sequence.…”

Section: ò ø óò 24º ([6]) a K-extended Skolem Sequence Of Order N Ismentioning

confidence: 99%

“…Ò Ø ÓÒ 2.12º ( [91]) If k ≤ n, a near Skolem sequence of order n and defect k is a k-near Skolem-type sequence of order n with m = 2n − 2. This sequence is also called the k-near Skolem sequence of order n. Ò Ø ÓÒ 2.13º ( [93]) When k ≤ n, a k-near Rosa sequence of order n is a k-near Skolem-type sequence of order n with m = 2n − 1 and s n = 0.…”

Section: ò ø óò 24º ([6]) a K-extended Skolem Sequence Of Order N Ismentioning

confidence: 99%

“…For example, if we concatenate a Skolem sequence of order n and a [hooked] Langford sequence of defect n + 2 and length l, we get a [hooked] (n + 1)-near Skolem sequence of order n + l + 1 (cf. [91]). …”

Section: Techniques Used In the Proofs Of The Existence Theoremsmentioning

confidence: 99%

“…Hooking (1, 1, 2, 0, 2) with L, we get (1, 1, 2, 5, 2, 11, 9, 7, 5, 12, 10, 8,6,4,7,9,11,4,6,8,10,12), which is a 3-near Skolem sequence of order 12 (cf. [91]). …”

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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