Disjoint skolem sequences and related disjoint structures

Abstract

A Skolem sequence of order n is a sequence S = (sl, s 2 . . . , szn) of 2n integers satisfying the following conditions: (1) for every k E {1, 2 , . . . ,n} there exist exactly two elements si,sj suchIn this article we show the existence of disjoint Skolem, disjoint hooked Skolem, and disjoint near-Skolem sequences. Then we apply these concepts to the existence problems of disjoint cyclic Steiner and Mendelsohn triple systems and the existence of disjoint 1-covering designs. 0 1993 John Wiley & Sons, Inc. SEQUENCES AND CYCLIC STSA Steiner triple system of order v, STS(v), is a pair of sets ( V , B ) , where IVl = v and B consists of 3-subsets (triples or blocks) of V such that any two elements of V occur in exactly one triple. An STS(v) exists if and only if v = 1 or 3 (mod 6). Two Steiner triple systems on the same set V are disjoint if they have no blocks in common. In [8,9,10,20], it is shown that the maximum number of painvise disjoint Steiner triple systems on a v-element set is v -2 for all v > 7. The maximum is 2 for v = 7.A STS(v) is cyclic if its automorphism group contains a v-cyclic. A cyclic STS (v) exists for all v = 1 or 3 (mod 6) except for v = 9 [13]. This question of existence is equivalent to finding solutions to Heffter's difference problems 161: * Research partially supported by grants from NSERC Canada.

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

“…By considering an exhaustive list of cases and constructing required sequences for each of them, L i n e k and J i a n g [49] proved that these conditions are also sufficient when the defect d is 2 or 3, except for d = 3 and (l, k) = (3, 2), (3,6), (4,1), (4,5), (4,9) when no such sequence exists. In the same paper, they also settled the existence problem of hooked k-extended Langford sequences of defect 2.…”

“…The idea of looped Langford sequences goes back to P r i d a y [74], and it found its application in construction of extended near-Skolem sequences. For example, (2,4,2,7,5,4,8,6,3,5,7,3,0,6,8), (2,4,2,7,5,4,6,8,3,5,7,3,6,0,0,8) is a looped Langford set of defect 2 and order 8.…”

“…Position the first element of RHS at the place of a hook in the HS, and put the last element of the HS at the place of the hook in the RHS. For example, L = (12,10,8,6,4,11,9,7,4,6,8,10,12,5,7,9,11,0,5) is a hooked Langford sequence of defect 4 and length 9. 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.…”

“…We will illustrate it on an example from a construction in [53]. The sequence (6,7,3,4,5,3,6,4,7,5) is a Langford sequence of defect 3 and length 5. We can pivot the second copy of integer 3 to the beginning of the sequence to get (3, 6, 7, 3, 4, 5, 0, 6, 4, 7, 5), a 7-extended Langford sequence of defect 3 and length 5.…”

“…The second definition was studied by N i c k e r s o n [64], and N o w a k o w s k i [66] proved that these two definitions are equivalent. For example, if n = 4, the partition {(2, 3), (6,8), (4,7), (1,5)}, or equivalently, the sequence (4, 1, 1, 3, 4, 2, 3, 2), is a (pure) Skolem sequence of order 4. On the other hand, the partition { (5,6), (1,3), (8,11), (9,13), (2,7), (4, 10)}, or in equivalent notation the sequence (2, 5, 2, 6, 1, 1, 5, 3, 4, 6, 3, 0, 4), is a hooked Skolem sequence of order 6. In order to prove the existence of cyclic Steiner triple systems, R o s a [79] introduced a split Skolem sequence of order n which is an (n + 1)-extended Skolem sequence of order n. Also, a hooked split Skolem sequence of order n is a hooked (n + 1)-extended Skolem sequence of order n. In more recent literature, these sequences are also called Rosa and hooked Rosa sequences of order n.…”