Numbers of common weights for extended triple systems

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Counting Configurations in Designs

AbstractGiven a t-(v, k, λ) design, form all of the subsets of the set of blocks. Partition this collection of configurations according to isomorphism and consider the cardinalities of the resulting isomorphism classes. Generalizing previous results for regular graphs and Steiner triple systems, we give linear equations relating these cardinalities. For any fixed choice of t and k, the coefficients in these equations can be expressed as functions of v and λ and so depend only on the design's parameters, and not its structure. This provides a characterization of the elements of a generating set for m-line configurations of an arbitrary design

The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems, together with the Supplement

A Skolem sequence of order n is a sequence S_n=(s_{1},s_{2},...,s_{2n}) of 2n integers containing each of the integers 1,2,...,n exactly twice, such that two occurrences of the integer j in {1,2,...,n} are separated by exactly j-1 integers. We prove that the necessary conditions are sufficient for existence of two Skolem sequences of order n with 0,1,2,...,n-3 and n pairs in same positions. Further, we apply this result to the fine structure of cyclic two, three and four-fold triple systems, and also to the fine structure of lambda-fold directed triple systems and lambda-fold Mendelsohn triple systems. For a better understanding of the paper we added more details into a "Supplement".Comment: The Supplement for the paper "The intersection spectrum of Skolem sequences and its applications to lambda fold cyclic triple systems" is available here. It comes right after the paper itsel

Properties of Steiner triple systems of order 21

Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and two that avoid the prism. All systems contain the grid. None have a block intersection graph that is 3-existentially closed.Comment: 12 page

Resolvability of infinite designs

In this paper we examine the resolvability of infinite designs. We show that in stark contrast to the finite case, resolvability for infinite designs is fairly commonplace. We prove that every t-(v,k,Λ) design with t finite, v infinite and k,λ<v is resolvable and, in fact, has α orthogonal resolutions for each α<v. We also show that, while a t-(v,k,Λ) design with t and λ finite, v infinite and k=v may or may not have a resolution, any resolution of such a design must have v parallel classes containing v blocks and at most λ−1 parallel classes containing fewer than v blocks. Further, a resolution into parallel classes of any specified sizes obeying these conditions is realisable in some design. When k<v and λ=v and when k=v and λ is infinite, we give various examples of resolvable and non-resolvable t-(v,k,Λ) designs

Disjoint skolem-type sequences and applications

Let D = {i₁, i₂,..., in} be a set of n positive integers. A Skolem-type sequence of order n is a sequence of i such that every i ∈ D appears exactly twice in the sequence at position aᵢ and bᵢ, and |bᵢ - aᵢ| = i. These sequences might contain empty positions, which are filled with 0 elements and called hooks. For example, (2; 4; 2; 0; 3; 4; 0; 3) is a Skolem-type sequence of order n = 3, D = f2; 3; 4g and two hooks. If D = f1; 2; 3; 4g we have (1; 1; 4; 2; 3; 2; 4; 3), which is a Skolem-type sequence of order 4 and zero hooks, or a Skolem sequence. In this thesis we introduce additional disjoint Skolem-type sequences of order n such as disjoint (hooked) near-Skolem sequences and (hooked) Langford sequences. We present several tables of constructions that are disjoint with known constructions and prove that our constructions yield Skolem-type sequences. We also discuss the necessity and sufficiency for the existence of Skolem-type sequences of order n where n is positive integers

Configurations and colouring problems in block designs

A Steiner triple system of order v (STS(v)) is called x-chromatic if x is the smallest number of colours needed to avoid monochromatic blocks. Amongst our results on colour class structures we show that every STS (19) is 3- or 4-chromatic, that every 3-chromatic STS(19) has an equitable 3-colouring (meaning that the colours are as uniformly distributed as possible), and that for all admissible v > 25 there exists a 3-chromatic STS(v) which does not admit an equitable 3-colouring. We obtain a formula for the number of independent sets in an STS(v) and use it to show that an STS(21) must contain eight independent points. This leads to a simple proof that every STS(21) is 3- or 4-chromatic. Substantially extending existing tabulations, we provide an enumeration of STS trades of up to 12 blocks, and as an application we show that any pair of STS(15)s must be 3-1-isomorphic. We prove a general theorem that enables us to obtain formulae for the frequencies of occurrence of configurations in triple systems. Some of these are used in our proof that for v > 25 no STS(u) has a 3-existentially closed block intersection graph. Of specific interest in connection with a conjecture of Erdos are 6-sparse and perfect Steiner triple systems, characterized by the avoidance of specific configurations. We describe two direct constructions that produce 6-sparse STS(v)s and we give a recursive construction that preserves 6-sparseness. Also we settle an old question concerning the occurrence of perfect block transitive Steiner triple systems. Finally, we consider Steiner 5(2,4, v) designs that are built from collections of Steiner triple systems. We solve a longstanding problem by constructing such systems with v = 61 (Zoe’s design) and v = 100 (the design of the century)

Algorithms for classification of combinatorial objects

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one game in every round. Such a characterization is called a classification for the objects of interest. Classification is typically conducted up to a notion of structural equivalence (isomorphism) between the objects. For example, one can view two tournament schedules as having the same structure if one can be obtained from the other by renaming the teams and reordering the rounds. This thesis examines algorithms for classification of combinatorial objects up to isomorphism. The thesis consists of five articles – each devoted to a specific family of objects – together with a summary surveying related research and emphasizing the underlying common concepts and techniques, such as backtrack search, isomorphism (viewed through group actions), symmetry, isomorph rejection, and computing isomorphism. From an algorithmic viewpoint the focus of the thesis is practical, with interest on algorithms that perform well in practice and yield new classification results; theoretical properties such as the asymptotic resource usage of the algorithms are not considered. The main result of this thesis is a classification of the Steiner triple systems of order 19. The other results obtained include the nonexistence of a resolvable 2-(15, 5, 4) design, a classification of the one-factorizations of k-regular graphs of order 12 for k ≤ 6 and k = 10, 11, a classification of the near-resolutions of 2-(13, 4, 3) designs together with the associated thirteen-player whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.reviewe