40 research outputs found
Combinatorics of the Sonnet
Using a definition of a sonnet, the number of basic rhyming schemes is enumerated. This is then used to discuss the 86 sonnets which appear in John Clare\u27s The Rural Muse
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Completing the design spectra for graphs with six vertices and eight edges
Apart from two possible exceptions, the design spectrum has been determined for every graph with six vertices and at most eight edges. The purpose of this note is to establish the existence of the two missing designs, both of order 32
Theta Graph Designs
We solve the design spectrum problem for all theta graphs with 10, 11, 12, 13, 14 and 15 edges
Subcubic trades in Steiner triple systems
We consider the problem of classifying trades in Steiner triple systems such that each block of the trade contains one of three fixed elements. We show that the fundamental building blocks for such trades are 3-regular graphs that are 1-factorisable. In the process we also generate all possible 2- and 3-way simultaneous edge colourings of graphs with maximum degree 3 using at most 3 colours, where multiple edges but not loops are allowed. Moreover, we generate all possible Latin trades within three rows
Further biembeddings of twofold triple systems
We construct face two-colourable triangulations of the graph 2Kn in an orientable surface; equivalently biembeddings of two twofold triple systems of order n, for all n ξ 16 or 28 (mod 48). The biembeddings come from index 1 current graphs lifted under a group ℤn/4 × 4
Distributive and anti-distributive Mendelsohn triple systems
We prove that the existence spectrum of Mendelsohn triple systems whose
associated quasigroups satisfy distributivity corresponds to the Loeschian
numbers, and provide some enumeration results. We do this by considering a
description of the quasigroups in terms of commutative Moufang loops.
In addition we provide constructions of Mendelsohn quasigroups that fail
distributivity for as many combinations of elements as possible.
These systems are analogues of Hall triple systems and anti-mitre Steiner
triple systems respectively
On the upper embedding of Steiner triple systems and Latin squares
It is proved that for any prescribed orientation of the triples of either a Steiner triple system or a Latin square of odd order, there exists an embedding in an orientable surface with the triples forming triangular faces and one extra large face
Types of directed triple systems
We introduce three types of directed triple systems. Two of these, Mendelsohn directed triple systems and Latin directed triple systems, have previously appeared in the literature but we prove further results about them. The third type, which we call skewed directed triple systems, is new and we determine the existence spectrum to be v ≡ 1 (mod 3), v ≠ 7, except possibly for v = 22, as well as giving enumeration results for small orders