Small embeddings of partial directed triple systems and partial triple systems with even λ

• Published in: Journal of combinatorial theory. Series A Vol. 37; no. 3; pp. 363 - 369
• Main Authors: Colbourn, Charles J, Hamm, Rose C, Rodger, C.A
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.01.1984; Academic Press
• Subjects: Combinatorics; Combinatorics. Ordered structures; Designs and configurations; Exact sciences and technology; Mathematics; Sciences and techniques of general use; Combinatorics
• ISSN: 0097-3165; 1096-0899
• DOI: 10.1016/0097-3165(84)90059-1

In 1976, Lindner and Rosa ( Ars Combin. 1 (1976) , 159–166) showed that a partial triple system with λ > 1 can be embedded in a finite triple system with the same λ. This result is improved in the case when λ is even by embedding a partial triple system on υ symbols in a triple system on t symbols, t ≡ 0,1 (mod 3), for all t >/ 3( λυ 2 + υ(2 − λ) + 1). In the process, it is shown that for any λ >/ 1, a partial directed triple system on υ symbols can be embedded in a directed triple system on t symbols, t ≡ 0, 1 (mod 3), for all t ⩾ 6 λv 2 + 6 v(1 − λ) + 3, thus generalizing a result of Hamm (Proceedings, 14th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1983).