Strong Skolem Starters

Abstract

This paper concerns a class of combinatorial objects called Skolem starters, and more specifically, strong Skolem starters, which are generated by Skolem sequences. In 1991, Shalaby conjectured that any additive group ${\mathbb{Z}}_n$, where $n\equiv 1$ or $3(\mathrm{mod}8),n\ge 11$, admits a strong Skolem starter and constructed these starters of all admissible orders $11\le n\le 57$. Only finitely many strong Skolem starters have been known to date. In this paper, we offer a geometrical interpretation of strong Skolem starters and explicitly construct infinite families of them.

17 pages, 2 figures