New proofs are given of the fundamental results of Bader, Lunardon and Thas relating flocks of the quadratic cone in PG(3, q), q odd, and BLT-sets of Q(4, q). We also show that there is a unique BLT-set of H(3, 9). The model of Penttila for Q(4, q), q odd, is extended to Q(2m, q) to construct partial flocks of size qm/2+m/2 – 1 of the cone kin PG(2m – 1, q) with vertex a point and base Q(2m – 2, q), where q is congruent to 1 or 3 modulo 8 and m is even. These partial flocks are larger than the largest previously known for m > 2. Also, the example of O'Keefe and Thas of a partial flock of k in PG(5, 3) of size 6 is generalised to a partial flock of the cone k of PG(2pn – 1, p) of size 2pn, for any prime p congruent to 1 or 3 modulo 8, with the corresponding partial BLT-set of Q(2pn, p) admitting the symmetric group of degree 2pn + 1.

We find necessary and sufficient conditions for completing an arbitrary 2 by n latin rectangle to an n by n symmetric latin square, for completing an arbitrary 2 by n latin rectangle to an n by n unipotent symmetric latin square, and for completing an arbitrary 1 by n latin rectangle to an n by n idempotent symmetric latin square. Equivalently, we prove necessary and sufficient conditions for the existence of an (n−1)-edge colouring of Kn (n even), and for n-edge colouring of Kn (n odd) in which the colours assigned to the edges incident with two vertices are specified in advance.

An n-hedral tiling of ℝd is a tiling with each tile congruent to one of the n distinct sets. In this paper, we use the iterated function systems (IFS) to generate n-hedral tilings of ℝd. Each tile in the tiling is similar to the attractor of the IFS. These tiles are fractals and their boundaries have the Hausdorff dimension less than d. Our results generalize a result of Bandt.

A planar graph contains faces which can be classified into types depending on the number of edges on the face boundaries. Under various natural rules for randomly dividing faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

In 1960, Trevor Evans gave a best possible embedding of a partial latin square of order n in a latin square of order t, for any t ≥ 2n. A latin square of order n is equivalent to a 3-cycle system of Kn, n, n, the complete tripartite graph. Here we consider a small embedding of partial 3k-cycle systems of Kn, n, n of a certain type which generalizes Evans' Theorem, and discuss how this relates to the embedding of patterned holes, another recent generalization of Evans' Theorem.

A balanced directed cycle design with parameters (υ, k, 1), sometimes called a (υ, k, 1)-design, is a decomposition of the complete directed graph into edge disjoint directed cycles of length k. A complete classification is given of (υ, k, 1)-designs admitting the holomorph {øa, b: x ↦ ax + b∣ a, b ∈ Zυ, (a, υ1) = 1} of the cyclic group Zυ as a group of automorphisms. In particular it is shown that such a design exists if and ony if one of (a) k = 2, (b) p ≡ 1 (mod k) for each prime p dividing υ, or (c) k is the least prime dividing υ, k2 does not divide υ, and p ≡ 1 (mod k) for each prime p < k dividing υ.

Consider a forest of maple trees in autumn, with leaves falling on the ground. Those coming late cover the others below, so eventually the fallen leaves form a statistically homogeneous spatial pattern. In particular, the uncovered leaf boundaries form a mosaic. We formulate a mathematical model to describe this mosaic, firstly in the case where the leaves are polygonal and later for leaves with curved boundaries. Mean values of certain statistics of the mosaic are derived.

Three differently defined classes of two-symbol sequences, which we call the two-distance sequences, the linear sequences and the characteristic sequences, have been discussed by a number of authors and some equivalences between them are known. We present a self-contained proof that the three classes are the same (when ambiguous cases of linear sequences are suitably in terpreted). Associated with each sequence is a real invariant (having a different appropriate definition for each of the three classes). We give results on the relation between sequences with the same invariant and on the symmetry of the sequences. The sequences are closely related to Beatty sequences and occur as digitized straight lines and quasicrystals. They also provide examples of minimal word proliferation in formal languages.

A complementary decomposition of λKn into a graph G is an edge-disjoint decomposition of λKn into copies of G such that if each copy H of G is replaced by its complement in V(H) then the result is an edge-disjoint decomposition of λKn into copies of GC it is a self- complementary decomposition if G = Gc. The spectrum for the last self-complementary graph on at most 7 vertices is found.

Let S be a finite linear space on v ≥ n2 –n points and b = n2+n+1–m lines, m ≧ 0, n ≧ 1, such that at most m points are not on n + 1 lines. If m ≧ 1, except if m = 1 and a unique point on n lines is on no line with two points, then S embeds uniquely in a projective plane of order n or is one exceptional case if n =4. If m ≦ 1 and if v ≧ n2 – 2√n + 3, + 6, the same conclusion holds, except possibly for the uniqueness.

1991 Mathematics subject classification (Amer. Math. Soc.) 05 B 05, 51 E 10.

It is well-known that if G is a multigraph (that is, a graph with multiple edges), the maximum number of pairwise disjoint edges in G is ν(G) and its maximum degree is D(G), then |E(G)| ≤ ν [3D/2’. We extend this theorem for r-graphs (that is, families of r-element sets) and for r-multihypergraphs (that is, r-graphs with repeated edges). Several problems remain open.

A balanced tournament design, BTD(n), defined on a 2n—set V is an arrangement of the () distinct unordered pairs of the elements of V into an n × 2n − 1 array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most two cells of each row. In this paper, we investigate the existence of balanced tournament designs with a pair of almost orthogonal resolutions. These designs can be used to construct doubly resolvable (ν, 3, 2)- BIBD s and, in our smallest applications, have been used to construct previously unknown doubly resolvable (ν, 3, 2)- B I B D s.

A construction for balanced ternary designs is given. Based on the designs so obtained, a construction of partially balanced ternary designs is given, which gives balanced ternary designs and series of symmetric balanced ternary designs in special cases.

The k-profile of an Hadamard matrix of order n is a function defined on the integers 0, 1,…,n. If k is even, k-profile of an Hadamard matrix of order n (k even) has non-zero terms only in every eighth position. If k is divisible by 4, the non-zero positions are those congruent to n (modulo 8).

We describe several recursive constructions for designs which use designs with “holes”. As an application, we give a short new proof of the Doyen-Wilson Theorem.

An infinite class of T-matrices is constructed using Golay sequences. A list is given with new Hadamard matrices of order 2t. q, q odd, q < 10000, improving the known values of t.

Finally T-matrices are given of order 2m + 1, for small values of m ≤ 12 which do not coincide with those generated by Turyn sequences.

The purpose of this paper is to prove (1) if q ≡ 1 (mod 8) is a prime power and there exists a Hadamard matrix of order (q − 1)/2, then we can construct a Hadamard matrix of order 4q, (2) if q ≡ 5 (mod 8) is a prime power and there exists a skew-Hadamard matrix of order (q + 3)/2, then we can construct a Hadamard matrix of order 4(q + 2), (3) if q ≡ 1 (mod 8) is a prime power and there exists a symmetric C-matrix of order (q + 3)/2, then we can construct a Hadamard matrix of order 4(q + 2).

We have 36, 36 and 8 new orders 4n for n ≤ 10000, of Hadamard matrices from the first, the second and third theorem respectively, which were known to the list of Geramita and Seberry. We prove these theorems by using an adaptation of generalized quaternion type array and relative Gauss sums.

In this note, we use a geometric construction to produce group divisible designs with mutually orhogonal resolutions.

A Kirkman square with index λ, latinicity μ, block size k and ν points, KSk(v; μ, λ), is a t × t array (t = λ(ν−1)/μ(k − 1)) defined on a ν-set V such that (1) each point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a (ν, k, λ)-BIBD. For μ = 1, the existence of a KSk(ν; μ, λ) is equivalent to the existence of a doubly resolvable (ν, k, λ)-BIBD. In this case the only complete results are for k = 2. The case k = 3, λ = 1 appears to be quite difficult although some existence results are available. For k = 3, λ = 2 the problem seems to be more tractable. In this paper we prove the existence of a KS3(ν; 1, 2) for all ν ≡ 3 (mod 12).

A direct construction for partially resolvable t-partitions is presented and then used to give a recursive construction for BIBDs (ν, 4, 2). In particular, we construct BIBD(ν, 4, 2) with BIBD(ν, 4, 2) embedded in it whenever ν = 3u + a, a ∈ {1, 4, 7}. This result allows us to give simple proofs for the existence of BIBD(ν, 4, 2) with various additioinal properties.