On Products of Strong Skolem Starters

TL;DR

This paper introduces a new method for generating strong Skolem starters of composite order using product operations on existing 2-partitions, expanding the known families beyond cardioidal constructions.

Contribution

It defines multi-valued binary operations to produce new infinite families of strong Skolem starters from existing ones, including non-cardioidal examples.

Findings

New product operations generate infinite families of strong Skolem starters.

Operations preserve properties like strong, skew, and cardioidal in 2-partitions.

Extended the class of known Skolem starters beyond previous constructions.

Abstract

In 1991, Shalaby conjectured that any $\mathbb{Z}_{n}$, where $n\equiv 1$ or $3\pmod{8},\ n\ge 11$, admits a strong Skolem starter. In 2018, the authors explicitly constructed some infinite "cardioidal" families of strong Skolem starters. No other infinite families of these combinatorial designs were known to date. Statements regarding the products of starters, proven in this paper give a new way of generating strong or skew Skolem starters of composite orders. This approach extends our previous result by generating new infinite families that are not cardioidal. The products that we introduce in this paper are multi-valued binary operations which produce 2-partitions of the set $\mathbb{Z}^*_{nm}$ of integers modulo $nm$ without zero, from a pair of 2-partitions of $\mathbb{Z}^*_n$ and $\mathbb{Z}^*_m$, where $n, m\ge 3$ are odd integers. We prove several remarkable properties of these operations applied to starters in $\mathbb{Z}_n$ and to some other combinatorial objects that are 2-partitions of $\mathbb{Z}^*_n$ with additional restrictions, such as strong, skew, Skolem and cardioidal 2-partitions.