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 such
In 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.
1. SEQUENCES AND CYCLIC STS
A 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.