Strong Skolem Starters

TL;DR

This paper explores strong Skolem starters, providing a geometric interpretation and constructing infinite families, advancing understanding of these combinatorial objects and addressing longstanding conjectures.

Contribution

It introduces a geometric perspective on strong Skolem starters and explicitly constructs infinite families, expanding the known classes of these objects.

Findings

Provided a geometric interpretation of strong Skolem starters.

Constructed infinite families of strong Skolem starters.

Addressed the conjecture on the existence of strong Skolem starters for certain groups.

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 $n\equiv1$ or $3\pmod{8},\ n\ge11$, admits a strong Skolem starter and constructed these starters of all admissible orders $11\le n\le57$. 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.