Alexander Rosa

One‐factorizations of the complete graph—A survey

We survey known results on the existence and enumeration of many kinds of 1-factorizations of the complete graph. We also mention briefly some related questions and topics, as well as applications.

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&

Steiner triple systems with rotational automorphisms

A Steiner triple system S(@u) of order @u is said to be k-ratational (k positive integer) if it admits an automorphism consisting of a single fixed point and precisely k (@u-1)@?k-cycles. We obtain a necessary and sufficient condition for the existence of 1-rotational and 2-rotational Steiner triple systems. We also enumerate nonisomorphic 1- and 2-rotational S(@u)s of small orders.

An Updated Bibliography and Survey of Steiner Systems

Two earlier versions of our bibliography on Steiner systems [D44], [D44.1] listed 400 and 675 titles, respectively. The number of papers concerning Steiner systems continues to grow, and the present listing contains already some 730 titles; all additions are recent papers. As for inclusion into our bibliography of papers dealing with finite projective, affine or inversive spaces, we follow the same policy as in [D44] and [D44.1] Information about reviews in referative journals is attached to the titles in the bibliography; MR stands for Mathematical Reviews while Z stands for Zentral-blatt fur Mathematik und ihre Grenzgebiete.

Quadratic leaves of maximal partial triple systems

Every graph having vertex degrees zero and two satisfying the basic necessary conditions is the leave of a maximal partial triple system, with one exception (C4 ⋃C5). The proof technique is direct, using the method of differences.

Cyclic One-Factorization of the Complete Graph

It is shown that the complete graph K n has a cyclic 1-factorization if and only if n is even and n ≠2 t , t ⩾3.

A Theorem on the Maximum Number of Disjoint Steiner Triple Systems

We prove that D(2v + 1) > v + 1 + D(v) for v > 3 where D(v) denotes the maximum number of pairwise disjoint Steiner triple systems of order v. Since D(v) Q u - 2 it follows that for v > 3, D(2v + 1) = 2v - 1 whenever D(v) = v - 2. A Steiner triple system (briefly STS) in a pair (S, B) where S is a set and B is a collection of 3-subsets of S (called triples) such that every 2-subset of S is contained in exactly one triple of B. The number 1 S 1 is called the order of the STS (S, B). It is well known that there is an STS of order u if and only if a E 1 or 3 (mod 6). Therefore in saying that a certain property concerning STS is true for all u it is understood that v E 1 or 3 (mod 6). An STS of order v will sometimes be denoted by STS(v). Two Steiner triple systems (S, BJ and (S, B2) are said to be disjoint if Bl and B, have no triples in common. We denote by D(U) the maximum number of pairwise disjoint STS(v); it is easy to show that D(v) d o - 2 for v 3 3 [4]. Following [9] we call any set of v - 2 pairwise disjoint STS(v) a large set of disjoint STS(u).

Another class of balanced graph designs: balanced circuit designs

A block considered as a set of elements together with its adjacency matrix A is called a C-block if A is the adjacency matrix of a circuit. A balanced circuit design with parameters v, b, r, k, @l (briefly, BCD(v, k, @l)) is an arrangement of v elements into bC-blocks such that each C-block contains k elements, each element occurs in exactly rC-blocks and any two distinct elements are linked in exactly @l C-blocks. We investigate conditions for the existence of BCD and show, in particular, that if the block-size k =< 6 and the trivial necessary conditions are satisfied, then the corresponding BCD exists.

Hanani triple systems

Hanani triple systems onv≡1 (mod 6) elements are Steiner triple systems having (v−1)/2 pairwise disjoint almost parallel classes (sets of pairwise disjoint triples that spanv−1 elements), and the remaining triples form a partial parallel class. Hanani triple systems are one natural analogue of the Kirkman triple systems onv≡3 (mod 6) elements, which form the solution of the celebrated Kirkman schoolgirl problem. We prove that a Hanani triple system exists for allv≡1 (mod 6) except forv ∈ {7, 13}.

On reverse Steiner triple systems

The existence of reverse Steiner triple systems (i.e. Steiner triple systems with a given involutory automorphism of special type) is investigated. It is shown that such a system exists for all orders n if n @? 1 or 3 or 9 (mod 24) except possibly for n = 25. A system with this property exists also for n = 19 and possibly for every n @? 19 (mod 24). On the other hand, it is demonstrated that such systems do not exist for the other values of n.