Disjoint skolem sequences and related disjoint structures

Abstract A Skolem sequence of order n is a sequence S = (s 1, s 2 …, s 2n) of 2 n integers satisfying the following conditions: (1) for every k ∈ 1, 2, … n there exist exactly two elements s i, S j such that S i = S j = k ; (2) If s i = s j = k, i < j then j − i = k. 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. © 1993 John Wiley & Sons, Inc.