Eric Mendelsohn

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.

New recursive methods for transversal covers

A transversal cover is a set of gk points in k disjoint groups of size g and a minimum collection of transversal subset s, called blocks, such that any pair of points not contained in the same group appear in at least one block. The case g = 2 was investigated and completely solved by Sperner, Renyi, Katona, Kleitman, and Spencer. For all g, asymptotic results are known, but little is understood for small values of k. Sloane and others have initiated the investigation of g = 3. The present article is concerned with constructive techniques for all g and k. One of the principal constructions generalizes Wilsons theorem for transversal designs. This article also discusses a simulated annealing algorithm for finding transversal covers and gives a table of the best known transversal covers at this time.

Lower Bounds for Transversal Covers

A transversal cover is a set of gk points in k disjoint groups of size g and a collection of b transversal subsets, called blocks, such that any pair of points not contained in the same group appears in at least one block. A central question is to determine, for given g, the minimum possible b for fixed k, or, alternatively, the maximum k for fixed b. The case g=2 was investigated and completely solved by Sperner sperner:28, Rényi renyi:71, Katona katona:73, and Kleitman and Spencer kleitman:73. For arbitrary g, asymptotic results are known but little is understood for small values of k. Constructions exist but these only produce upper bounds on b. The present article is concerned with lower bounds on b. We develop three general lower bounds on b for fixedg and k. The first one is proved using one of the principal constructions brett:97a, the second comes from the study of intersecting set-systems, and the third is shown by a set packing argument. In addition, we investigate upper bounds on k for small fixed b. This proves useful to reduce or eliminate the gap between lower and upper bounds on b for some transversal covers with small k.

On the Groups of Automorphisms of Steiner Triple and Quadruple Systems

In [4] Jean Doyen raised the question of whether every group was the auto- morphism group of a Steiner Triple System (STS). The answer to this is trivially in the affirmative using techniques of Frucht, Hedrlin, Pultr, and others (cf. [lo]) but there is a serious defect in that even for finite groups the systems obtained are infinite. The purpose of this paper is to correct this defect and extend the result to Steiner quadruple systems (SQS). The result is obtained through methods of universal algebra as well as combinatorial methods. As this paper will deal with both the cases of STS’s and SQS’s and both cases are similar but with distinct differences we shall adopt the notation Theorem l(T), Theorem l(Q), etc., whenever the essential nature of the argument is the same but the differences between STS’s and SQS’s is sufficient to bear noting, e.g.,

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)

Anti-mitre steiner triple systems

A mitre in a Steiner triple system is a set of five triples on seven points, in which two are disjoint. Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least 9/16 of the admissible orders. Computational results for small cyclic Steiner triple systems are also included.

Orthogonal latin square graphs

An orthogonal latin square graph (OLSG) is one in which the vertices are latin squares of the same order and on the same symbols, and two vertices are adjacent if and only if the latin squares are orthogonal. If G is an arbitrary finite graph, we say that G is realizable as an OLSG if there is an OLSG isomorphic to G. The spectrum of G [Spec(G)] is defined as the set of all integers n that there is a realization of G by latin squares of order n. The two basic theorems proved here are (1) every graph is realizable and (2) for any graph G, Spec G contains all but a finite set of integers. A number of examples are given that point to a number of wide open questions. An example of such a question is how to classify the graphs for which a given n lies in the spectrum.

On Partially Resolvable t–Partitions

Publisher Summary This chapter discusses partially resolvable t -partitions. The chapter discusses primarily with the case |P|= 1, |S| = 1. The corresponding partially resolvable t -partitions are denoted by PRP t -( p, s, υ ; m ) provided P = { p }, S = { s }. These designs are interesting on their own and they also arise naturally in many instances, for example, when considering embeddings of Steiner triple systems, Doyen and Wilson, recursive constructions for Steiner quadruple systems, Lindner and Rosa, and other occasions. Necessary conditions for the existence of PRP t-( p, s, υ; m ) are discussed and some general results and constructions are presented. The chapter also discusses the existence of PRP t-( p, s, υ; m ) in the two smallest non-trivial cases, that is, when t = 2, p = 2, s = 3, and t = 2, p = 3, s = 2, respectively. The case of PRP 2-(2, 4, υ; m ) is discussed. Two examples of partially resolvable 3-partitions are given in the chapter.

Covering arrays avoiding forbidden edges

Covering arrays (CAs) can be used to detect the existence of faulty pairwise interactions between parameters or components in a software system. The generalization considered here applies to the situation in which some input combinations are invalid, a requirement quite common in software testing. In this paper, we study covering arrays avoiding forbidden edges (CAFEs), where certain pairwise interactions are forbidden while all others must be covered, and we aim to minimize the number of tests. We establish a theoretical framework for this problem, by providing connections to the edge clique covering problem, lower and upper bounds, complexity results and a recursive construction. We also give an algorithm for the case of binary alphabets.

A survey of Skolem-type sequences and Rosa’s use of them

Let D be a set of positive integers. A Skolem-type sequence is a sequence of i ∈ D such that every i ∈ D appears exactly twice in the sequence at positions ai and bi, and |bi − ai| = i. These sequences might contain empty positions, which are filled with null elements. Thoralf A. Skolem defined and studied Skolem sequences in order to generate solutions to Heffter’s difference problems. Later, Skolem sequences were generalized in many ways to suit constructions of different combinatorial designs. Alexander Rosa made the use of these generalizations into a fine art. Here we give a survey of Skolem-type sequences and their applications.