In the paper we study the asymptotic behaviour of the number of trees with n vertices and diameter k = k(n), where k/√n → ∞ as n → ∞ but k = o(n).

Commutative idempotent quasigroups with a sharply transitive automorphism group G are described in terms of so-called Room maps of G. Orthogonal Room maps and skew Room maps are used to construct Room squares and skew Room squares. Very general direct and recursive constructions for skew Room maps lead to the existence of skew Room maps of groups of order prime to 30. Also some nonexistence results are proved.

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

An equidistant permutation array is a ν × r array A(r, λ;ν) defined on a r-set X such that every row of A is a permutation of X and any two distinct rows agree in precisely λ common columns. Define In this paper, we show that where n = r − λ. Certain results pertaining to irreducible equidistant permutation arrays are also established.

We show that the problem of settling the existence of an n × n Hadamard matrix, where n is divisible by 4, is equivalent to that of finding the cardinality of a smallest set T of 4-circuits in the complete bipartite graph K n, n, such that T contains at least one circuit of each copy of K2,3 in Kn, n.

The (2, 3, ν) bipacking number is determined for all integers ν, and the number of non-isomorphic bipackings is found for small values of ν. The general solution for lambada packings of pairs into triples is deduced from the results for λ = 1 and λ = 2.

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.

It is shown that if an interval graph possesses a maximal-clique partition then its clique covering and clique partition numbers are equal, and equal to the maximal-clique partition number. Moreover an interval graph has such a partition if and only if all its maximal cliques are edge-disjoint.

We prove Theorem 1: suppose G is a simple graph of order n having Δ(G) = n − k where k ≥ 5 and n ≥ max (13, 3k −3). If G contains an independent set of k − 3 vertices, then the TCC (Total Colouring Conjecture) is true. Applying Theorem 1, we also prove that the TCC is true for any simple graph G of order n having Δ(G) = n −5. The latter result together with some earlier results confirm that the TCC is true for all simple graphs whose maximum degree is at most four and for all simple graphs of order n having maximum degree at least n − 5.

A new, elementary proof of the Macdonald identities for An−1 using induction on n is given. Specifically, the Macdonald identity for An is deduced by multiplying the Macdonald identity for An−1 and n Jacobi triple product identities together.

Some sufficient conditions for the reconstructability of separable graphs are given proceeding along the lines suggested by Bondy, Greenwell and Hemminger. It is shown that the structure and automorphism group of a central block plays an important role in the reconstruction.

We give a short proof that skew Room squares exist for all odd sides s exceeding 5.

A graph H decomposes into a graph G if one can write H as an edge-disjoint union of graphs isomorphic to G. H decomposes into D, where D is a family of graphs, when H can be written as a union of graphs each isomorphic to some member of D, and every member of D is represented at least once. In this paper it is shown that the d-dimensional cube Qd decomposes into any graph G of size d each of whose blocks is either an even cycle or an edge. Furthennore, Qd decomposes into D, where D is any set of six trees of size d.

It is known that 4 ≤ x(ℝ2) ≤ 7, where x(ℝ2) is the number of colour necessary to colour each point of Euclidean 2-space so that no two points lying distance 1 apart have the same colour. Any lattice-sublattice colouring sucheme for R2 must use at least 7 colour to have an excluded distance. This article shows that at least 6 colours are necessary for an excluded distance when convex polygonal tiles (all with area greater than some positive constant) are used as the colouring base.

An infinite family of vertex-and edge-transitive, but not arc-transitive, graphs of degree 4 is constructed.

Pfaffian graphs are those which can be oriented so that the 1-factors have equal sign, as calculated according to the prescription of Kasteleyn. We consider various operations on graphs and examine their effect on the Pfaffian property. We show that the study of Pfaffian graphs may be reduced to the case of subcubic graphs (graphs in which no vertex has degree greater than 3) or bricks (3-connected bicritical graphs).

It is known that the problem of settling the existence of an n × n Hadamard matrix, where n is divisible by 4, is equivalent to that of finding the cardinality of a smallest set T of 4-circuits in the complete bipartite graph Kn, n such that T contains at least one circuit of each copy of K2,3 in Kn, n. Here we investigate the case where n ≡ 2 (mod 4), and we show that the problem of finding the cardinality of T is equivalent to that of settling the existence of a certain kind of n × n matrix. Moreover, we show that the case where n ≡ 2 (mod 4) differs from that where n ≡ 0 (mod 4) in that the problem of finding the cardinality of T is not equivalent to that of maximising the determinant of an n × n (1,-1)-matrix.

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 Latin square is considered to be a set of n2 cells with three block systems. An automorphisni is a permutation of the cells which preserves each block system. The automorphism group of a Latin Square necessarily has at least 4 orbits on unordered pairs of cells if n < 2. It is shown that there are exactly 4 orbits if and only if the square is the composition table of an elementary abelian 2-group or the cclic group of order 3.

Vertices u and v of a graph G are pseudo-similar if G – u ≅ G – v, but no automorphisms of G maps u to v. Let H be a graph with a distinguished vertex a. Denote by G(u. H) and G(v. H) the graphs obtained from G and H by identifying vertex a of H with pseudo-similar vertices u and v, respectively, of G. Is it possible for G(u.H) and G(v.H) to be isomorphic graphs? We answer this question in the affirmative by constructing graphs G for which G(u. H)≅ G(v. H).