On the number of transversals in a class of Latin squares

Denote by $\mathcal{A}_p^k$ the Latin square of order $n=p^k$ formed by the Cayley table of the additive group $(\mathbb{Z}_p^k,+)$, where $p$ is an odd prime and $k$ is a positive integer. It is shown that for each $p$ there exists $Q>0$ such that for all sufficiently large $k$, the number of transversals in $\mathcal{A}_p^k$ exceeds $(nQ)^{\frac{n}{p(p-1)}}$

Hamiltonian embeddings from triangulations

A Hamiltonian embedding of Kn is an embedding of Kn in a surface, which may be orientable or non-orientable, in such a way that the bound-ary of each face is a Hamiltonian cycle. Ellingham and Stephens recently established the existence of such embeddings in non-orientable surfaces for n = 4 and n ≥ 6. Here we present an entirely new construction which produces Hamiltonian embeddings of Kn from triangulations of Kn when n ≡ 0 or 1 (mod 3). We then use this construction to obtain exponential lower bounds for the numbers of nonisomorphic Hamiltonian embeddings of Kn

Maximum genus embeddings of Steiner triple systems

We prove that for n>3 every STS(n) has both an orientable and a nonorientable embedding in which the triples of the STS(n) appear as triangular faces and there is just one additional large face. We also obtain detailed results about the possible automorphisms of such embeddings

Wilson-Schreiber colourings of cubic graphs

We classify all spherically symmetric and homothetic spacetimes that are allowed kinematically by constructing them from a small number of building blocks. We then restrict attention to a particular dynamics, namely perfect fluid matter with the scale-free barotropic equation of state p=?? where 0&lt;~?&lt;1 is a constant. We assign conformal diagrams to all solutions in the complete classification of Carr and Coley, and so establish which of the kinematic possibilities are realized for these dynamics. We pay particular attention to those solutions which arise as critical solutions during gravitational collapse.<br/

Properties of the Steiner Triple Systems of Order 19

Properties of the 11 084 874 829 Steiner triple systems of order 19 are examined. In particular, there is exactly one 5-sparse, but no 6-sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number 3; all have chromatic index 10, except for 4 075 designs with chromatic index 11 and two with chromatic index 12; all are 3-resolvable; and there are exactly two 3-existentially closed STS(19)

A flaw in the use of minimal defining sets for secret sharing schemes

It is shown that in some cases it is possible to reconstruct a block design D uniquely from incomplete knowledge of a minimal defining set for D. This surprising result has implications for the use of minimal defining sets in secret sharing schemes

On independent sets

In a general set-theoretic context, an independent set is defined as a set which avoids certain specified structures called blocks. A formula is given for the number of independent sets of cardinality $k$ in terms of the numbers of configurations (i.e. non-empty collections) of blocks

On maximal partial Latin hypercubes

A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension $d$ and order $n$. The result generalises and extends previous results for $d=2$ (Latin squares) and $d=3$ (Latin cubes). Explicit constructions show that this bound is near-optimal for large $n> d$. For $d>n$, a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least