Further 6-sparse Steiner Triple Systems (original) (raw)
Related papers
Properties of the Steiner Triple Systems of Order 19
The Electronic Journal of Combinatorics, 2010
Properties of the 11$\,$084$\,$874$\,$829 Steiner triple systems of order 19 are examined. In particular, there is exactly one 5-sparse, but no 6-sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number 3; all have chromatic index 10, except for 4$\,$075 designs with chromatic index 11 and two with chromatic index 12; all are 3-resolvable; and there are exactly two 3-existentially closed STS(19).
On sparse countably infinite Steiner triple systems
Journal of Combinatorial Designs, 2009
We give a general construction for Steiner triple systems on a countably infinite point set and show that it yields 2 ℵ 0 nonisomorphic systems all of which are uniform and r-sparse for all finite r ≥ 4. q
Sets of three pairwise orthogonal Steiner triple systems
Journal of Combinatorial Theory, Series A, 2003
Two Steiner triple systems (STS) are orthogonal if their sets of triples are disjoint, and two disjoint pairs of points defining intersecting triples in one system fail to do so in the other. In 1994, it was shown (Canad. J. Math. 46(2) (1994) 239-252) that there exist a pair of orthogonal Steiner triple systems of order v for all v 1; 3 (mod 6), with vX7; va9: In this paper we show that there exist three pairwise orthogonal Steiner triple systems of order v for all v 1 ðmod 6Þ; with vX19 and for all v 3 ðmod 6Þ; with vX27 with only 24 possible exceptions. r
On the Szamkolowicz-Doyen Classification of Steiner Triple Systems
Proceedings of the London Mathematical Society, 1977
1. Introduction A Steiner triple system (8, stf) is a finite set 8 together with a family s/ of triples (subsets of cardinality 3) such that each pair of elements of S is contained in exactly one triple. The elements of 8 are called points, the elements of s/ are called lines, and three points which are not on a line are a triangle. The cardinal of 8, which we denote by 18 |, is the order of the Steiner triple system (S,s/). It is well known [8] that a necessary and sufficient condition for the existence of a Steiner triple system of order n > 0 is that n = 1 or 3 (mod 6). Trivial Steiner triple systems occur when S = stf -0, when 8 consists of one point and s& = 0, and when 8 consists of three points and $2 consists of one line. A subsystem of a Steiner triple system (S, stf) is a Steiner triple system (8', &4') such that 8' <= 8 and s#' <= s0. It is easy to see that if (8 V J^) , ..., (8 r ,s0 r ) are subsystems of a Steiner triple system then so is (8 x n... n8 r ,s^n... ns/ r ). It follows that given a set P s S of points, there is a unique minimal subsystem containing P. This subsystem is said to be generated by P.
Towards a large set of Steiner quadruple systems
SIAM Journal on Discrete Mathematics, 1991
SIAM i-LUST. Math, Й 1991 Society for Industrial and Applied Mathematics Vol.4, No. 2, pp. 182-195, May 19!) I 004 TOWARDS A LARGE SET OF STEINER QUADRUPLE SYSTEMS* TUVI ETZIONf and ALAN HARTMANf Abstract. Let D(u) be the number of pairwise disjoint ...
The Fine Intersection Problem for Steiner Triple Systems
Graphs and Combinatorics, 2008
The intersection of two Steiner triple systems (X, A) and (X, B) is the set A ∩ B. The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m, n) such that there exist two Steiner triple systems of order v whose intersection I satisfies | ∪A∈I A| = m and |I| = n. We show that for v ≡ 1 or 3 (mod 6), |I(v)| = Θ(v 3), where previous results only imply that |I(v)| = Ω(v 2).
Journal of Combinatorial Designs, 2005
In this paper, we present a conjecture that is a common generalization of the Doyen-Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v v 1, 3 ðmod 6Þ, u < v v < 2u þ 1, we ask for the minimum r such that there exists a Steiner triple system ðU, BÞ, jUj ¼ u such that some partial system ðU, Bnq qÞ can be completed to an STSðv vÞ, ðV, B 0 Þ, where jq qj ¼ r. In other words, in order to ''quasi-embed'' an STSðuÞ into an STSðv vÞ, we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity ðuðu À 1Þ=6Þ À r as the maximum intersection of an STSðuÞ and an STSðv vÞ with u < v v. We conjecture that the necessary minimum r ¼ ðv v À uÞð2u þ 1 À v vÞ=6 can be achieved, except when u ¼ 6t þ 1 and v v ¼ 6t þ 3, in which case it is r ¼ 3t for t 6 ¼ 2, or r ¼ 7 when t ¼ 2. Using small examples and recursion, we solve the cases v v À u ¼ 2 and 4, asymptotically solve the cases v v À u ¼ 6, 8, and 10, and further show for given v v À u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v v À u). Some results are obtained for v v close to 2u þ 1 as well. The cases where v v % 3u=2 seem to be the hardest.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.