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

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

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

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

Independent sets in Steiner triple systems

A set of points in a Steiner triple system $({\rm STS}(v))$ is said to be independent if no three of these points occur in the same block. In this paper we derive for each $k\le8$ a closed formula for the number of independent sets of cardinality $k$ in an ${\rm STS}(v)$. We use the formula to prove that every STS(21) has an independent set of cardinality eight and is as a consequence 4-colourable

Biembeddings of Latin squares and Hamiltonian decompositions

Face 2-colourable triangulations of complete tripartite graphs $K_{n,n,n}$ correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for $n=3,4,5$ and 6, and we summarize the corresponding results for $n=7$. Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs $K^*_{n,n}$, and we also give computational results for these in the cases $n=3,4,5$ 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