52 research outputs found
On biembedding an idempotent latin square with its transpose
Let L be an idempotent Latin square of side n, thought of as a set of ordered triples (i, j, k) where L(I, j) = k. Let I be the set of triples (i, I, i). We consider the problem of biembedding the triples of L\I, with the triples of L'\ I, where L' is the transpose of L, in an orientable surface. We construct such embeddings for all doubly even values of n
Open data from the third observing run of LIGO, Virgo, KAGRA, and GEO
The global network of gravitational-wave observatories now includes five detectors, namely LIGO Hanford, LIGO Livingston, Virgo, KAGRA, and GEO 600. These detectors collected data during their third observing run, O3, composed of three phases: O3a starting in 2019 April and lasting six months, O3b starting in 2019 November and lasting five months, and O3GK starting in 2020 April and lasting two weeks. In this paper we describe these data and various other science products that can be freely accessed through the Gravitational Wave Open Science Center at https://gwosc.org. The main data set, consisting of the gravitational-wave strain time series that contains the astrophysical signals, is released together with supporting data useful for their analysis and documentation, tutorials, as well as analysis software packages
Configurations and trades in Steiner triple systems
The main result of this paper is the determination of all pairwise nonisomorphic
trade sets of volume at most 10 which can appear in Steiner
triple systems. We also enumerate partial Steiner triple systems having
at most 10 blocks as well as configurations with no points of degree 1 and
tradeable configurations having at most 12 blocks.
AMS classification: 05B0
The Triangle chromatic index of Steiner triple systems
In a Steiner triple system of order v, STS(v), a set of three lines intersecting pairwise in three distinct points is called a triangle. A set of lines containing no triangle is called triangle-free. The minimum number of triangle-free sets required to partition the lines of a Steiner triple system S, is called the triangle chromatic index of S. We prove that for all admissible v, there exists an STS (v) with triangle chromatic index at most 8√3v. In addition, by showing that the projective geometry PG(n,3) may be partitioned into O(6n/5) caps, we prove that the STS(v) formed the points and lines of the affine geometry AG(n,3) has triangle chromatic index at most Avs, where s=log6/(3log5)≈0.326186, and A is a constant. We also determine the values of the index for STS(v) with v≤13
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Orientable self-embeddings of Steiner triple systems of order 15
It is shown that for 78 of the 80 isomorphism classes of Steiner triple systems of order 15 it is possible to find a face 2-colourable triangulation of the complete graph K_{15} in an orientable surface in which the colour classes both form representatives of the specified isomorphism class. For one of the two remaining isomorphism classes it is proved that this is not possible. We also discuss the remaining open case
On 6-sparse Steiner triple systems
We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in
the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction
which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also
made concerning existing construction methods for perfect Steiner triple systems, and we give a further
example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a
uniform Steiner triple system of order 180,907
Biembeddings of Latin squares and Hamiltonian decompositions
Face 2-colourable triangulations of complete tripartite graphs correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for and 6, and we summarize the corresponding results for . Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs , and we also give computational results for these in the cases and 6
Modular gracious labellings of trees
A gracious labelling g of a tree is a graceful labelling in which, treating the tree as a bipartite graph, the label of any edge (d,u) (d a 'down' and u an 'up' vertex) is g(u) - g(d). A gracious k-labelling is one such that each residue class modulo k has teh 'correct' numbers of vertex and edge labels -- that is, the numbers that arise by interpreting the labels of a gracious labelling modulo k. In this paper it is shown that every non-null tree has a gracious k-labelling for each k = 2,3,4,5
On colourings of Steiner triple systems
A Steiner triple system, STS(v), is said to be χ-chromatic if the points can be coloured using χ colours, but no fewer, such that no block is monochromatic. All known 3-chromatic STS(v) are also equitably colourable, i.e. there exists a 3-colouring in which the cardinalities of the colour classes differ by at most one. We present examples of 3-chromatic STS(v) which do not admit equitable 3-colourings. We also present further examples of systems with unique and balanced colouring
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