Mathematical and algorithmic methods for finding disjoint Rosa-type sequences

Alruhaymi, Fatimah (2018) Mathematical and algorithmic methods for finding disjoint Rosa-type sequences. Masters thesis, Memorial University of Newfoundland.

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Abstract

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 CTS₄(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 CTS₄(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.

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