zbMATH — the first resource for mathematics

Generating sets in Steiner triple systems. (English) Zbl 0984.05013
Let \(W\) be a subset of the point set \(V\) of a Steiner triple system (STS). The set \(W\) is a \(k\)-generating set, if in the Steiner quasigroup associated with the STS, every element of \(V\) can be written as a product of at most \(k\) elements of \(W\). When \(k=2\) such a generating set is a spanning or dominating set, and these have been applied in constructions of STS with complete arcs. The authors study the case \(k=3\) under the condition that every element of \(V\setminus W\) can be written in exactly one way as a product of at most 3 elements of \(W\).

05B07 Triple systems
Full Text: EuDML
[1] ASSAF A. M.: Modified group divisible designs. Ars Combin. 29 (1990), 13-20. · Zbl 0702.05014
[2] BIGELOW D. C.-COLBOURN, C J.: Faithful enclosings of triple systems: A gen ralization of Stern’s theorem. Graphs, Matrices, and Designs (R. Rees, Dekker, New York, NY, 1992, pp. 31-42. · Zbl 0792.05027
[3] BRANDES M. DE-RÖDL V.: Steiner triple systems with small maximal independent sets. Ars Combin. 17 (1984), 15-19. · Zbl 0548.51010
[4] BROWN T. C.-BUHLER J. P.: A density version of a geometric Ramsey theorem. J. Combin. Theory Ser. A 32 (1982), 20-34. · Zbl 0476.51008
[5] COLBOURN, C J.-DINITZ J. H.-STINSON D. R.: Spanning sets and scattering sets in Steiner triple systems. J. Combin. Theoru Ser. A 57 (1991), 46-59. · Zbl 0765.05016
[6] COLBOURN, C J.-HOFFMAN D. G.-REES R.: A new class of group divisible designs with block size three. J. Combin. Theoru Ser. A 59 (1992), 73-89. · Zbl 0759.05012
[7] COLBOURN, C J.-ROSA A.: Quadratic leaves of maximal partial triple systems. Graphs Combin. 2 (1986), 317-337. · Zbl 0609.05009
[8] COLBOURN, C J.-ROSA A.: Triple Systems. Oxford Universitu Press, Oxford, 1999. · Zbl 0938.05009
[9] ERDÖS P.-HAJNAL A.: On chromatic number of graphs and set systems. Acta Math. Acad. Sci. Hung. 17 (1966), 61-99. · Zbl 0151.33701
[10] GRABLE D. A.-PHELPS K. T.-RODL V.: The minimum independence number for designs. Combinatorica 15 (1995), 175-185. · Zbl 0824.05005
[11] HANANI H.: Balanced incomplete block designs and related designs. Discrete Math. 11 (1975), 255-369. · Zbl 0361.62067
[12] MIAO Y.-ZHU L.: Existence of incomplete group divisible designs. J. Combin. Math. Combin. Comput. 6 (1989), 33-49. · Zbl 0688.05009
[13] PHELPS K. T.-RÖDL V.: Steiner triple systems with minimum independence number. Ars Combin. 21 (1986), 167-172. · Zbl 0625.05012
[14] SAUER N.-SCHÖNHEIM J.: Maximal subsets of a given set having no triple in common with a Steiner triple system on the set. Canad. Math. Bull. 12 (1969), 777-778. · Zbl 0194.01002
[15] WEI R.: Group-divisible designs with equal-sized holes. Ars Combin. 35 (1993), 315-324. · Zbl 0786.05012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.