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Mathematical and algorithmic methods for finding disjoint Rosa-type sequences
A Rosa sequence of order n is a sequence S = (s1; s2; ..., s2n+1) of 2n + 1 integers
satisfying the conditions: (1) for every k ∈ {1; 2;...; n} there are exactly two elements
sᵢ; sj ∈ S such that si = sj = k; (2) if sᵢ = sj = k; i < j, then j - i = k; and (3)
sn+1 = 0 (sn+1 is called the hook). Two Rosa sequences S and S' are disjoint if
sᵢ = sj = k = s't = s'ᵤ implies that {i;j} ≠{t,u}, for all k = 1;..., n.
In 2014, Linek, Mor, and Shalaby [18] introduced several new constructions for
Skolem, hooked Skolem, and Rosa rectangles.
In this thesis, we gave new constructions for four mutually disjoint hooked Rosa
sequences and we used them to generate cyclic triple systems CTSâ‚„(v). We also obtained
new constructions for two disjoint m-fold Skolem sequences, two disjoint m-fold
Rosa sequences, and two disjoint indecomposable 2-fold Rosa sequences of order n.
Again, we can use these sequences to construct cyclic 2-fold 3-group divisible design
3-GDD and disjoint cyclically indecomposable CTSâ‚„(6n+3). Finally, we introduced
exhaustive search algorithms to find all distinct hooked Rosa sequences, as well as
maximal and maximum disjoint subsets of (hooked) Rosa sequences