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

The Existence of Two New Classes of Near-Skolem Type Sequences

Let m,n be two positive integers, m ≤ n. An m-near-Skolem sequence of order n and defect m is a sequence m − Sn = (s1, s2, · · · , s2n−2) of 2n − 2 non-negative integers such that the following conditions hold: 1. for each k ∈ {1, 2, · · · ,m−1,m+1, · · · , n} there exist exactly two elements si, sj ∈ m − Sn such that si = sj = k 2. if si = sj = k, then |i − j| = k For example, (4, 1, 1, 2, 4, ...

Indecomposable Skolem and Rosa sequences

We introduce indecomposable (hooked) Skolem sequences and we show their existence for all admissible orders. We then introduce a new construction for two-fold triple systems from two-fold Skolem sequences, and use this to motivate the concept of a two-fold Rosa sequence, whose existence we show for all admissible orders.

NP-completeness of generalized multi-Skolem sequences

A Skolem sequence is a sequence a1, a2, . . . , a2n (where ai ∈ A = {1, . . . , n}), each ai occurs exactly twice in the sequence and the two occurrences are exactly ai positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The existence question of deciding which sets of the form A = {1, . . . , n} are Skolem sets was solved by Thoralf Skolem [6] in 195...

Skolem arrays and skolem labellings of ladder graphs

Extending Skolem sequences, how can you begin?

For each n ≥ 1 and 1 ≤ j ≤ n we show the existence of an extended Skolem sequence of order n starting with the symbol j.