New constructions of strong and Skolem starters

Abstract

This thesis studies a class of combinatorial objects called strong starters, and their subclass, strong Skolem starters, which are generated by Skolem sequences. In 1991, Shalaby conjectured that the additive group ℤₙ, for any 𝑛 ≡ 1 or 3 (mod 8), 𝑛 ≥ 11, admits a strong Skolem starter and constructed these starters of all admissible orders 11 ≤ 𝑛 ≤ 57. Only finitely many strong Skolem starters have been known prior to the discovery of infinite families of them in this thesis. A geometrical interpretation of strong Skolem starters and explicit construction of the infinite cardioidal family of strong Skolem starters are offered in the thesis. Then by employment of operations called products of starters, a new way of generating strong Skolem starters of composite orders is received. This approach extends the previous result by generating new infinite families of strong Skolem starters that are not cardioidal. This discovery of the infinite families of strong Skolem starters gives significant support to Shalaby’s conjecture. Finally, a process of triplication of given strong starters (in ℤₚ, where 3 ∤ 𝑝) that yields strong starters in ℤ₃ₚ and ℤ₉ₚ is thoroughly studied. The study results in the method that allows us to construct such strong starters in ℤ₃ₚ for 7 ≤ 𝑝 ≤ 49, 𝑝 coprime to 6, and in ℤ₉ₚ, 𝑝 = 5, 7, 11, 13, 17, by hand, which is otherwise quite a challenging task. This research sheds a new light on Horton’s conjecture stating that strong starters exist in any additive abelian group of odd orders except ℤ₃, ℤ₅, ℤ₃ + ℤ₃ and ℤ₉.