Disjoint skolem-type sequences and applications

Let D = {i₁, i₂,..., in} be a set of n positive integers. A Skolem-type sequence of order n is a sequence of i such that every i ∈ D appears exactly twice in the sequence at position aᵢ and bᵢ, and |bᵢ - aᵢ| = i. These sequences might contain empty positions, which are filled with 0 elements and called hooks. For example, (2; 4; 2; 0; 3; 4; 0; 3) is a Skolem-type sequence of order n = 3, D = f2; 3; 4g and two hooks. If D = f1; 2; 3; 4g we have (1; 1; 4; 2; 3; 2; 4; 3), which is a Skolem-type sequence of order 4 and zero hooks, or a Skolem sequence. In this thesis we introduce additional disjoint Skolem-type sequences of order n such as disjoint (hooked) near-Skolem sequences and (hooked) Langford sequences. We present several tables of constructions that are disjoint with known constructions and prove that our constructions yield Skolem-type sequences. We also discuss the necessity and sufficiency for the existence of Skolem-type sequences of order n where n is positive integers

On the number of additive permutations and Skolem-type sequences

Cavenagh and Wanless recently proved that, for sufficiently large odd n, the number of transversals in the Latin square formed from the addition table for integers modulo n is greater than (3.246)n. We adapt their proof to show that for sufficiently large t the number of additive permutations on [-t,t] is greater than (3.246)2t+1 and we go on to derive some much improved lower bounds on the numbers of Skolem-type sequences. For example, it is shown that for sufficiently large t ≡ 0$ or 3 (mod 4), the number of split Skolem sequences of order n=7t+3 is greater than (3.246)6t+3. This compares with the previous best bound of 2⌊n/3⌋

Starter sequences: generalizations and applications

In this thesis we introduce new types of starter sequences, pseudo-starter sequences, starter-labellings, and generalized (extended) starter sequences. We apply these new sequences to graph labeling. All the necessary conditions for the existence of starter, pseudo-starter, extended, m-fold, excess, and generalized (extended) starter sequences are determined, and some of these conditions are shown to be sufficient. The relationship between starter sequences and graph labellings is introduced. Moreover, the starter-labeling and the minimum hooked starter-labeling of paths, cycles, and k- windmills are investigated. We show that all paths, cycles, and k-windmills can be starter-labelled or minimum starter-labelled

A generalization of the problem of Mariusz Meszka

Mariusz Meszka has conjectured that given a prime p=2n+1 and a list L containing n positive integers not exceeding n there exists a near 1-factor in K_p whose list of edge-lengths is L. In this paper we propose a generalization of this problem to the case in which p is an odd integer not necessarily prime. In particular, we give a necessary condition for the existence of such a near 1-factor for any odd integer p. We show that this condition is also sufficient for any list L whose underlying set S has size 1, 2, or n. Then we prove that the conjecture is true if S={1,2,t} for any positive integer t not coprime with the order p of the complete graph. Also, we give partial results when t and p are coprime. Finally, we present a complete solution for t<12.Comment: 15 page

The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems, together with the Supplement

A Skolem sequence of order n is a sequence S_n=(s_{1},s_{2},...,s_{2n}) of 2n integers containing each of the integers 1,2,...,n exactly twice, such that two occurrences of the integer j in {1,2,...,n} are separated by exactly j-1 integers. We prove that the necessary conditions are sufficient for existence of two Skolem sequences of order n with 0,1,2,...,n-3 and n pairs in same positions. Further, we apply this result to the fine structure of cyclic two, three and four-fold triple systems, and also to the fine structure of lambda-fold directed triple systems and lambda-fold Mendelsohn triple systems. For a better understanding of the paper we added more details into a "Supplement".Comment: The Supplement for the paper "The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems" is available here. It comes right after the paper itsel