On Products of Strong Skolem Starters

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.