Charles C. Lindner

Steiner quadruple systems - a survey☆

A Steiner system S( t, k, O) is a pair (S, B) where S is a u-set and B is a collection of k-subsets of S such that every t-subset of S is contained in exactly one member of B. A system S(2,3, u) is called a Steiaer triple system (briefly STS) and a system S(3,4, u) is called a Steiner quadruple system (briefly SQS). Steiner systems S(t, k, v) were apparently defined for the first time by Woolhouse in 1844 [76] who asked: for which ir tegers t, k, u does an S(t, k, U) exist? This problem remains unsolved in general until today. However, several partial answers were given only 3 years later in 1847 by Kirkman [28] who showed that a system S(2,3, v) i(i.e. an STS) exists if and only if v = 1 or 3 (mod 6) and constructed systems S(3,4,2”) (i.e. SQS) for r=very n. (In today’s geometrical terminology, the elements and quadruples of such a system are respectively the points and planes of the n-dimensional af5ne space over GF(2).) It was not until several years later that Steiner [67] asked for the existence of systems S(t, t+ 1, v). The fact that systems S(t, k;, U) still carry the commonly accepted name “Steiner systems” s probably due to the fact that the papers [76] and [28] were completely overlooked by writt;rs on the subject in the late nineteenth and early twentieth century. During :5is time very much was written on the subject of STS and not much on SQS. III 1896, Moore, among other things, posed the problem of the existence of systems S(t, k, U) in [56]. In 1908, Barrau [4] established the uniquen :ss of S(3,4,8) and S(3,4, lo), 13nd in 1915, Fitting [163 constructed cyclic S(3,4,26) anI4 S(3,4,34). In 1935 Bzys and de Week [5] showed the existence of an S(3,4,14). Other work in the thirties relating directiy or indirectly to SQS include:s [8], [72] and [75]. But it was not until 1960 that Hangnl [24] proved that *he wcessary condition :, = 2 or 4 (-nod 6) for the existence of an S(3,4, ~1) is also su cient . Althou&

On the construction of odd cycle systems

Three obvious necessary conditions for the existence of a k-cycle system of order n are that if n > 1 then n ⩾ k, n is odd, and 2k divides n(n − 1). We show that if these necessary conditions are sufficient for all n satisfying k ⩽ n < 3k then they are sufficient for all n. In particular, there exists a 15-cycle system of order n if and only if n ≡ 1, 15, 21, or 25 (mod 30), and there exists a 21-cycle system of order n if and only if n ≡ 1, 7, 15, or 21 (mod 42), n ≠ 7. 15.

Steiner pentagon systems

A Steiner pentagon system is a pair (Kn, P) where Kn isthe complete undirected graph on n vertices. P is a collection of edge-disjoint pentagons which partition Kn, and such that every part of distinct vertices of Kn is joined by a path of length two in exactly one pentagon of the collection P. The number n is called the order of the system. This paper gives a somplete solution of the existence problem of Steiner pentagon systems. In particular it is shown that the spectrum for Steiner pentagon systems (=the set of all orders for which a Steiner pentagon system exists) is precisely the set of all n ? 1 or 5 (mod 10), except 15, for which no such system exists.

2-Perfect m -cycle systems

Abstract The spectrum for 2-perfect m -cycle systems of K n has been considered by several authors in the case when m ⩽7. In this paper we essentially solve the problem for 2-perfect m -cycle systems of K n in the case where m is prime and 2 m +1 is a prime power. In particular we settle the problem for m = 11 and 13 except for two or one possible exceptions respectively. The problem for m = 9 is also considered.

On the number of 1-factorizations of the complete graph

It is well known that for every positive integer IZ there exists a l-factorization of the complete graph KS,, . (For this result and for undefined graph-theoretical notions and standard notation, see [12].) Although the question about the existence of 1-factorizations of Kzn is answered easily, the problem of determining the number N(2n) of pairwise nonisomorphic I-factorizations of Kz, appears to be a difficult one. Known results on N(2n) can be summarized as follows: N(2) = N(4) = N(6) = 1 (this is easily obtained). Further, N(8) = 6 (proved by Safford [7] in 1906 and again by Wallis [18] in 1972). Gelling ([9]; see also [IO]) used a computer to obtain N(10) = 396 (he also determined the orders of the groups of the respective I-factorizations). Finally, a recent result of Wallis [19] states that N(2n) > 2 for n > 4. The main purpose of this paper is to improve this last result. We show in Section 3, among other things, that the number N(2n) goes to infinity with n, by making use of the relationship between I-factorizations and quasigroups satisfying certain identities (this relationship has apparently been noticed also in [13, 141). The same result is proved again in Section 5 where we use two recursive constructions to show that the number A(2n)

The spectrum for 2-perfect 6-cycle systems

Abstract Recently, the spectrum problem for 2-perfect m -cycle systems has been studied by several authors. In this paper we find the spectrum for 2-perfect 6-cycle systems with two possible exceptions. The connection between these systems and quasigroups satisfying some 2 variable identities is discussed.

On the spectrum of resolvable orthogonal arrays invariant under the Klein group K4

It is shown that there exists a resolvablen2 by 4 orthogonal array which is invariant under the Klein 4-groupK4 for all positive integersn congruent to 0 modulo 4 except possibly forn ∈ {12, 24, 156, 348}.

Perfect hexagon triple systems

Abstract The graph consisting of the three 3-cycles ( a , b , c ), ( c , d , e ), and ( e , f , a ), where a , b , c , d , e , and f are distinct is called a hexagon triple. The 3-cycle ( a , c , e ) is called an “inside” 3-cycle; and the 3-cycles ( a , b , c ), ( c , d , e ), and ( e , f , a ) are called “outside” 3-cycles. A 3 k -fold hexagon triple system of order n is a pair ( X , C ), where C is an edge disjoint collection of hexagon triples which partitions the edge set of 3 kK n . Note that the outside 3-cycles form a 3 k -fold triple system. If the hexagon triple system has the additional property that the collection of inside 3-cycles ( a , c , e ) is a k -fold triple system it is said to be perfect . A perfect maximum packing of 3 kK n with hexagon triples is a triple ( X , C , L ), where C is a collection of edge disjoint hexagon triples and L is a collection of 3-cycles such that the insides of the hexagon triples plus the inside of the triangles in L form a maximum packing of kK n with triangles. This paper gives a complete solution (modulo two possible exceptions) of the problem of constructing perfect maximum packings of 3 kK n with hexagon triples.

Perfect dexagon triple systems

We completely determine the spectrum for perfect dexagon triple systems.

The construction of large sets of indempotent quasigroups

The maximum number of idempotent quasigroups of order n which pairwise agree on the main diagonal only is n − 2. Such a collection is called a large set of idempotent quasigroups of order n. The main result in this paper is the construction of a large set of idempotent quasigroups of order n for every n ⩾ 3 except n = 6, for which no such collection exists, and n = 14 and 62. Additionally, the known spectrum for large sets of Mendelsohn quasigroups is improved.