Ben Barber, Stefan Glock, Daniela Kühn, Allan Lo, Richard Montgomery, Deryk Osthus In Edge decompositions of graphs with high minimum degree, Daniela Kühn, Allan Lo, Deryk Osthus and I proved that the edge sets of sufficiently dense graphs satisfying necessary divisibility conditions could be partitioned into copies of an arbitrary graph . This result has since …

# Tag: graphs

## Chromatic number of the plane

The unit distance graph on has edges between those pairs of points at Euclidean distance . The chromatic number of this graph lies between (by exhibiting a small subgraph on vertices with chromatic number ) and (by an explicit colouring based on a hexagonal tiling of the plane). Aubrey de Grey has just posted a …

## Random Structures and Algorithms 2017

A partial, chronologically ordered, list of talks I attended at RSA in Gniezno, Poland. Under construction until the set of things I can remember equals the set of things I’ve written about. Shagnik Das A family of subsets of that shatters a -set has at least elements. How many -sets can we shatter with a …

## Clique decompositions of multipartite graphs and completion of Latin squares

Ben Barber, Daniela Kühn, Allan Lo, Deryk Osthus and Amelia Taylor, Journal of Combinatorial Theory, Series A, Volume 151, October 2017, Pages 146–201 PDF A Latin square of order is an grid of cells, each of which contains one of distinct symbols, such that no symbol appears twice in any row or column. There is a natural …

## Matchings without Hall’s theorem

In practice matchings are found not by following the proof of Hall’s theorem but by starting with some matching and improving it by finding augmenting paths. Given a matching in a bipartite graph on vertex classes and , an augmenting path is a path from to such that ever other edge of is an edge of …

## Partition regularity and other combinatorial problems

This is the imaginative title of my PhD thesis. It contains four unrelated pieces of work. (I was warned off using this phrasing in the thesis itself, where the chapters are instead described as “self-contained”.) The first and most substantial concerns partition regularity. It is a coherent presentation of all of the material from Partition …

## Fractional clique decompositions of dense graphs and hypergraphs

Ben Barber, Daniela Kühn, Allan Lo, Richard Montgomery and Deryk Osthus, Journal of Combinatorial Theory, Series B, Volume 127, November 2017, Pages 148–186 PDF Together with Daniela Kühn, Allan Lo and Deryk Osthus I proved that for every graph there is a constant such that every “-divisible” graph on vertices with minimum degree at least has an …