Overlap Cycles for Steiner Quadruple Systems

TL;DR

This paper proves that for all orders v congruent to 2 or 4 mod 6 greater than 4, Steiner quadruple systems exist that admit a 1-overlap cycle, extending the concept of universal cycles.

Contribution

It demonstrates the existence of 1-overlap cycles in Steiner quadruple systems for all applicable orders using Hanani's constructions.

Findings

Existence of 1-overlap cycles in SQS(v) for v > 4, v ≡ 2, 4 mod 6

Extension of universal cycle concepts to Steiner quadruple systems

Application of Hanani's constructions to establish these cycles

Abstract

Steiner quadruple systems are set systems in which every triple is contained in a unique quadruple. It is will known that Steiner quadruple systems of order v, or SQS(v), exist if and only if v = 2, 4 mod 6. Universal cycles, introduced by Chung, Diaconis, and Graham in 1992, are a type of cyclic Gray code. Overlap cycles are generalizations of universal cycles that were introduced in 2010 by Godbole. Using Hanani's SQS constructions, we show that for every v = 2, 4 mod 6 with v > 4 there exists an SQS(v) that admits a 1-overlap cycle.