Disjoint skolem sequences and related disjoint structures

A Skolem sequence of order n is a sequence S = (s₁, s₂…, s²ⁿ) of 2n 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.