On strong Skolem starters

TL;DR

This paper introduces a simple construction method for strong Skolem starters and cardioidal starters in various cyclic groups, expanding the known classes of such combinatorial structures.

Contribution

It provides an alternative, straightforward construction technique for strong Skolem starters in specific cyclic groups, generalizing previous results.

Findings

Constructs strong Skolem starters for rac{q^n}{q\u2261 3 (mod 8)}

Extends constructions to rac{pq}{p,q (mod 8)} and rac{p^n}{p (mod 8)}

Simplifies existing methods for generating cardioidal starters

Abstract

In this note we present an alternative (simple) construction of cardioidal starters (strong and Skolem) for $\mathbb{Z}_{q^n}$, where $q\equiv3$ (mod 8) is an odd prime number and $n\geq1$ is an integer number; also for $\mathbb{Z}_{pq}$ and $\mathbb{Z}_{p^n}$, for infinitely many odd primes $p,q\equiv1$ (mod 8) and $n\geq1$ an integer number. This cardioidal starters can be obtained by a more general result given by Ogandzhanyants et al. [O. Ogandzhanyants, M. Kondratieva and N. Shalaby, \emph{Strong Skolem starters}, J. Combin. Des. {\bf 27} (2018), no. 1, 5--21].