Kirkman Triple Systems Of Order Research Articles

On the seven non-isomorphic solutions of the fifteen schoolgirl problem

In this paper we give a simple and effective tool to analyze a given Kirkman triple system of order 15 and determine which of the seven well-known non-isomorphic KTS(15)s it is isomorphic to.Our technique refines and improves the lacing of distinct parallel classes introduced by F. N. Cole, by means of the notion of residual triple defined by G. Falcone and the present author in a previous paper.Unlike Cole's original lacing scheme, our algorithm allows one to distinguish two KTS(15)s also in the harder case where the two systems have the same underlying Steiner triple system. In the special case where the common STS is #19, an alternative method is given by means of the 1-factorizations of the complete graph K8 associated to the two KTSs.Moreover, we present three new visual solutions to the schoolgirl problem, and we catalogue most of the classical (or interesting) solutions in the literature in terms of what KTS(15)s they are isomorphic to.This paper provides background on a classical topic, while shedding new light on the problem as well.

An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns

AbstractA Kirkman triple system of order , KTS, is a resolvable Steiner triple system on elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS which contain as a subdesign a Steiner triple system of order , an STS. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case to (at present) three possible exceptions. In addition, we obtain results for other cases of the form and also near . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.

Kirkman triple systems with subsystems

A Steiner triple system of order v, STS(v), together with a resolution of its blocks is called a Kirkman triple system of order v, KTS(v). A KTS(v) exists if and only if v≡3(mod6). The smallest order for which the KTS(v) have not been classified is v=21, which is also the smallest order for which the existence of a doubly resolvable STS(v) is open. Here, KTS(21) with STS(7) and STS(9) subsystems are classified, leading to more than 13 million KTS(21). In this process, systems missing from an earlier classification of KTS(21) with nontrivial automorphisms are encountered, so such a classification is redone.

Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19

A Hanani triple system of order 6 n + 1 , HATS ( 6 n + 1 ) , is a decomposition of the complete graph K 6 n + 1 into 3 n sets of 2 n disjoint triangles and one set of n disjoint triangles. A nearly Kirkman triple system of order 6 n , NKTS ( 6 n ) , is a decomposition of K 6 n − F into 3 n − 1 sets of 2 n disjoint triangles; here F is a one-factor of K 6 n . The Hanani triple systems of order 6 n + 1 and the nearly Kirkman triple systems of order 6 n can be classified using the classification of the Steiner triple systems of order 6 n + 1 . This is carried out here for n = 3 : There are 3787983639 isomorphism classes of HATS ( 19 ) s and 25328 isomorphism classes of NKTS ( 18 ) s. Several properties of the classified systems are tabulated. In particular, seven of the NKTS ( 18 ) s have orthogonal resolutions, and five of the HATS ( 19 ) s admit a pair of resolutions in which the almost parallel classes are orthogonal.

Another construction for large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection $$\{(X, \mathcal {B}_i) : 1 \leq i \leq v - 2\}$$ , where every $$(X, \mathcal {B}_i)$$ is a KTS(v) and all $$\mathcal {B}_i$$ form a partition of all triples on X. In this article, we give a new construction for LKTS(6v + 3) via OLKTS(2v + 1) with a special property and obtain new results for LKTS, that is there exists an LKTS(3v) for $$v = \prod_{i=1}^{p}(2q_i^{r_i}+1)\prod_{j=1}^{q}(4^{s_j}-1)$$ , where p, q ? 0, r i , s j ? 1, q i is a prime power and $$q_i \equiv 7$$ mod 12.

Some infinite families of large sets of Kirkman triple systems

AbstractA large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection {(X, Bi) : 1 ≤ i ≤ v − 2}, where every (X,Bi) is a KTS(v) and all Bi form a partition of all triples on X. Many researchers have studied the existence of LKTS(v) for a long time. In [13], the author introduced a concept—large set of generalized Kirkman systems (LGKS), which plays an important role in the discussion of LKTS. In this article, we give a new construction for LGKS and obtain some new results of LKTS, that is, there exists an LKTS(6u + 3) for u = qn, where n ≥ 1, q ≡ 7 (mod 12) and q is a prime power. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 202–212, 2008

Can Kirkman's Schoolgirls Walk Abreast by Forming Arithmetic Progressions?

A Kirkman triple system of order v, whose point set is consecutively numbered, is called smooth, if for each parallel class there is a number d, such that each triple of the parallel class forms an arithmetic progression modulo v with common difference d. It will be shown that smooth Kirkman triple systems of order v exist, if and only if v is a power of 3. 1 Smooth Kirkman triple systems The following problem, known as Kirkman’s schoolgirl problem, was introduced in 1850 by Reverend Thomas J. Kirkman (1806–1895) as “Query 6” on page 48 of the Ladies and Gentleman’s Diary [2] (see [3] for his statement of the general problem): Fifteen young ladies in a school walk out three abreast for seven days in succession; it is required to arrange them daily, so that no two walk twice abreast. The general setting of the problem is the following: One wants to find 3n+1 arrangements of 6n + 3 girls in rows of three, such that any two girls belong to the same row in exactly one arrangement. In modern terms of block designs, the generalized Kirkman’s schoolgirl problem is equivalent to finding a resolution of some balanced incomplete block design with block size 3: For positive integers b, v, r, k, λ with r = λ(v−1) k−1 and b = λv(v−1) k(k−1) , a (v, k, λ)-Balanced Incomplete Block Design BIBD consists of a finite set Z of v elements, called points, and b subsets Z1, Z2, . . . , Zb of Z, called blocks, such that the following hold: every point occurs in exactly r blocks, every block contains exactly k points, and every pair of points occurs together in exactly λ blocks. 2000 Mathematics Subject Classification: 05B07 51E10 05B05 05C35 05D99 Key-words: Kirkman’s schoolgirl problem, Kirkman triple systems, resolvable designs, arithmetic progressions, cyclic groups

Support sizes of threefold resolvable triple systems

AbstractLet SSR(v, 3) denote the set of all integer b* such that there exists a RTS(v, 3) with b* distinct triples. In this paper, we determine the set SSR(v, 3) for v ≡ 3 (mod 6) and v ≥ 3 with only five undecided cases. We establish that SSR(v, 3) = P(v, 3) for v ≡ 3 (mod 6), v ≥ 21 and v ≠ 33, 39 where P(v, 3) = {mv, mv + 4, mv + 6, mv + 7, …, 3mv} and mv, = v(v − 1)/6. As a by‐product, we remove the last two undecided cases for the intersection numbers of Kirkman triple system of order 27, this improves the known result provided in [2]. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 275–289, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10037

An improved product construction for large sets of Kirkman triple systems

It has been shown by Lei, in his recent paper, that there exists a large set of Kirkman triple systems of order uv ( LKTS(uv)) if there exist an LKTS( v), a TKTS( v) and an LR( u), where a TKTS( v) is a transitive Kirkman triple system of order v, and an LR( u) is a new kind of design introduced by Lei. In this paper, we improve this product construction by removing the condition “there exists a TKTS( v)”. Our main idea is to use transitive resolvable idempotent symmetric quasigroups instead of TKTS. As an application, we can combine the known results on LKTS and LR-designs to obtain the existence of an LKTS(3 n m(2·13 n 1 +1)⋯(2·13 n t +1)) for n⩾1, m∈{1,5,11,17,25,35,43,67,91,123}∪{2 2r+125 s+1 : r⩾0,s⩾0} , t⩾0 and n i⩾1 (i=1,…,t) .

Kirkman triple systems of order 21 with nontrivial automorphism group

There are 50,024 Kirkman triple systems of order 21 admitting an automorphism of order 2. There are 13,280 Kirkman triple systems of order 21 admitting an automorphism of order 3. Together with the 192 known systems and some simple exchange operations, this leads to a collection of 63,745 nonisomorphic Kirkman triple systems of order 21. This includes all KTS(21)s having a nontrivial automorphism group. None of these is doubly resolvable. Four are quadrilateral-free, providing the first examples of such a KTS(21).

Embeddings of nearly Kirkman triple systems

It is proved in this paper that for u ≡ v ≡ 0 (mod 6), v > 100 and u ≥ 4v, any nearly Kirkman triple system of order v can always be embedded in a nearly Kirkman triple system of order u. Embeddings of nearly Kirkman triple systems of order v < 100 are also discussed.

Intersections of Kirkman triple systems

Abstract It is proved in this paper that for any v≡9 ( mod 18) , v⩾45, there exists a pair of Kirkman triple systems of order v having exactly k triples in common if and only if k∈{0,1,…,tv−6,tv−4,tv}, with the possible exception k=tv−4.

The 1-rotational Kirkman triple systems of order 33

We establish that there are exactly 500 KTS(33)s admitting an automorphism group fixing one point and acting regularly on the remainder; 436 are over the cyclic group while 64 are over the dicyclic group. There are exactly 243 nonisomorphic STS(33)s underlying the above KTS(33)s; 211 are over the cyclic group while 32 are over the dicyclic group. This gives a significant improvement on the number of known KTS(33)s (at least 528 instead of at least 28).

On Kirkman triple systems of order 33

Twenty-eight non-isomorphic KTS(33) with an automorphism of order 11 are constructed from the 84 cyclic STS(33).

Transitive Steiner and Kirkman triple systems of order 27

There are 71 Steiner triple systems of order 27 whose automorphism groups are point-transitive, and there are 248 transitive Kirkman triple systems of order 27. Computational methods used to find these designs are outlined. The designs and some of their properties are presented.

Transitive Steiner and Kirkman Triple Systems of Order 27

Transitive Steiner and Kirkman Triple Systems of Order 27

Partitioning Steiner triple systems into complete arcs

For a Steiner triple system of order v to have a complete s-arc one must have s( s + 1)/2⩾ v with equality only if s = 1 or 2 mod 4. To partition a Steiner triplesystem of order s(s + 1) 2 into complete s-arcs, one must have s = 1 mod 4. In this paper wegive constructions of Steiner triple systems of order s(s + 1) 2 which can be partitioned into complete s-arcs for all s = 1 mod 4. For s = 1 or 5 mod 12, we construct cyclic Steiner triple systems having this property. For s = 9 mod 12 we use Kirkman triple systems of order s having one additional property to construct these Steiner triple systems. We further establish that Kirkman triple systems having this additional property exist at least for s = 9 mod 24 and s = 21 mod 120.

A class of Steiner triple systems of order 21 and associated Kirkman systems

We examine a class of Steiner triple systems or order 21 with an automorphism consisting of three disjoint cycles of length 7. We exhibit explicitly all members of this class: they number 95 including the 7 cyclic systems. We then examine resolvability of the obtained systems; only 6 of the 95 are resolvable yielding a total of 30 nonisomorphic Kirkman triple systems of order 21. We also list several invariants of the systems and investigate their further properties.