The existence of augmented resolvable Steiner quadruple systems

Abstract

An augmented Steiner quadruple system of order v is an ordered triple ( X , B , E ) , where ( X , B ) is an SQS ( v ) and E is the set of all 2-subsets of X . An augmented Steiner quadruple system ( X , B , E ) of order v is resolvable if B ∪ E can be partitioned into n = ( v − 1 ) ( v + 4 ) / 6 parts B ∪ E = P 1 | P 2 | ⋯ | P n such that each part P i is a partition of X . Hartman and Phelps in [A. Hartman, K.T. Phelps, Steiner quadruple systems, in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory, Wiley, New York, 1992, pp. 205–240] conjectured that there exists a resolvable augmented Steiner quadruple systems of order v for any positive integer v ≡ 2 or 10 (mod 12). In this paper, we show that the Hartman and Phelps conjecture is true.