1-Overlap Cycles For Maximal Packing Quadruple Systems

A 3-(n, 4, 1) packing design consists of an n-element set X and a collection of 4-element subsets of X, called blocks, such that every 3-element subset of X is contained in at most one block. The(X, B) is called maximum if there does not exist any 3-(n, 4, 1)packing design(X, A) with |A| > |B|, and shortly denoted by M P QS(n). Hanani,Brouwer, Bao and Ji have determined the existence of M P QS(n).1-overlap cycles require a set of strings to be ordered so that the last letter of one string is the first letter of the next. When n ≡ 2, 4(mod 6), Horan and Hurlber showed the existence of the 1-overlap cycles for Steiner quadruple systems in 2014 based on Hanani’s constructions of Steiner quadruple systems. In this paper, the existence of 1-overlap cycle for M P QS(n) is determined, we also give a new existence proof of 1-overlap cycle for SQS(n) based on Hartman’s constructions of SQSs.For every n ≡ 0(mod 6), by using Brouwer’s construction we show that there exists an M P QS(n) that admits a 1-overlap cycle. We also show that for every n ≡ 1, 3(mod 6)there exists an SQS(n + 1) and an M P QS(n) that admit a 1-overlap cycle on the basis of Hartman’s constructions. And we show that for every n ≡ 5(mod 6) there exists an M P QS(n) that admits a 1-overlap cycle, using Bao and Ji’s constructions for M P QSs.