### Studies in Combinatorics: 17

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This is joint work with Jonathan Noel. This proves a conjecture of Chalcraft.

Determining whether a graph has an F-decomposition is NP-complete, but it is much easier to find 'fractional' decompositions. I'll also mention an application to completion of partial mutually orthogonal latin squares. I will discuss a construction of finite 'geometric' random graphs motivated by the study of random walks on infinite groups.

This construction has connections to other topics, including the Poisson boundary and Sznitman's random interlacements which I will try to introduce in a gentle way. Compressions play an important role in our proofs. Erdos asked the following question: given a positive integer n, what is the largest integer k such that any set of n points in a plane, with no 4 on a line, contains k points no 3 of which are collinear? This solves a longstanding problem in the area of extremal graph theory.

In this talk, I will present a family of graphs that represent particle interactions in a field theory. They also represent Grassmannians. Using matroid technology, I answered the physically interesting question of which diagrams correspond to positive Grassmannians.

I also discuss open questions regarding defining a differential operator, combinatorially, on this family of graphs. Bonato and Janssen showed that there are some countable random geometric graphs with the same property, but that this depended on the underlying geometry i. They also gave a partial classification of which norms in two dimensions do give a unique graph. We give a complete classification of all norms in all finite dimensions. The proof mixes combinatorics, probability and the geometry of finite dimensional lattices. Motivated by a sampling problem arising in statistics, Diaconis, Graham and Holmes introduced a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph G.

Two basic questions about this Markov chain are: for which class of graphs is the switch chain ergodic, and for which is it rapidly mixing? I'll present a precise answer to the ergodicity question and close bounds on the mixing question. In particular, I'll give some rough indications of a proof that the mixing time of the switch chain is polynomial in the size of the graph G in the case that G is monotone a. The class of monotone graphs includes examples of interest in the statistical setting.

Let A and B be two partially ordered sets. In this talk, I will prove that for every finite partially ordered set P, there exists a constant c P such that if a subset F of the hypercube ordered by inclusion does not contain P strongly, then the LYM density of F is at most c P , where the LYM density of F is defined to be the sum of the densities of F within each level of the hypercube. Recently there has been a lot of chatter from various parties about generalising Conway's extraordinary construction of the small Mathieu groups from projective planes.

For some reason several of these people seem to think I should put my oar in and so here are some thoughts on these recent constructions along these lines and potential avenues for generalisations. The well-known colouring number of a graph is one more than the smallest k for which there exists an ordering of the vertices in which each vertex has at most k neighbours that come earlier in the order. The colouring number is a trivial upper bound for the chromatic number of the graph.

But it also contains structural information of the graph, for instance regarding the edge density of any subgraph. When instead of neighbours that are earlier in the order we consider vertices that are earlier and can be reached by a path of length at most r, we get the generalised colouring numbers.

Depending on where we want the internal vertices of such paths to appear in the order, we actually can define different types of those generalised colouring numbers. Generalised colouring numbers were introduced by Kierstead and Yang in , to study specific types of graph colourings. In the last couple of years it has been realised that the generalised colouring numbers are closely related to many structural properties of graphs such as tree-depth and tree-width.

In this talk we give an overview of the relations between the generalised colouring numbers and specific types of colourings of graphs; we discuss relations between those numbers and structural properties of graphs; and we discuss some new bounds on these numbers for specific classes of graphs such as planar and minor-closed graphs.

A theorem of Lafforgue asserts that a matroid whose polytope cannot be dissected into smaller matroid polytopes can have only finitely many representations over any field, and in particular no continuously deformable representation. We explain a proof using the machinery of valuated matroids, which we present by way of hyperfields following Matt Baker's recent work.

This classical problem has recently enjoyed a renewed interest due to the current attention the quantum information community is giving to its complex analogue. I will report on some new developments of the theory of equiangular lines in Euclidean spaces. Among other things, I will present a new construction using real mutually unbiased bases and improvements to two long standing upper bounds for equiangular lines in dimensions 14 and An alternating sign matrix ASM is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1.

It was conjectured by Mills, Robbins and Rumsey in that the number of ASMs of fixed size is given by a certain simple product formula. A relatively short proof of this conjecture was obtained by Kuperberg in , using connections with the six-vertex model of statistical mechanics. It was also conjectured by Robbins in the mid 's that the number of ASMs of fixed odd size which are invariant under diagonal and antidiagonal reflection is given by a simple product formula.

This conjecture has only recently been proved, again using connections with the six-vertex model, in my joint work with Ilse Fischer and Matjaz Konvalinka see arXiv In the second part, I'll outline the proof of the enumeration formula for diagonally and antidiagonally symmetric ASMs of odd order. Our tools include some nonabelian Fourier analysis and a Brascamp--Lieb-type inequality for the symmetric group due to Carlen, Lieb, and Loss.

Informally, a body-bar framework consists of some rigid bodies and some bars each connecting two different rigid bodies. We can use multigraphs to illustrate body-bar frameworks by adding a vertex for each rigid body and adding k edges between two vertices if there are k bars connecting the rigid bodies corresponding to these vertices.

Tay showed that a generic body-bar framework is rigid if and only if the underlying multigraph contains the union of six edge-disjoint spanning trees, where a framework is generic if the endpoints of the bars are algebraically independent over the rationals. We will summarise Tay's result and then introduce some classes of non-generic body-bar frameworks that have the same characterisation as the generic ones. Many of the fastest known algorithms for factoring large integers rely on finding subsequences of randomly generated sequences of integers whose product is a perfect square.

In a paper published in Annals of Mathematics in , Croot, Granville, Pemantle and Tetali significantly improved these bounds, and stated a conjecture as to the location of this sharp threshold. In recent work, we have confirmed this conjecture. In my talk, I shall give a brief overview of some of the ideas used in the proof, which relies on techniques from number theory, combinatorics and stochastic processes.

Classical linear inequality systems can be examined well by linear programming. The tropical counterpart is more difficult to handle. We examine the problem from a combinatorial point of view related to tropical oriented matroids. Furthermore, we show connections to scheduling and games. We'll present buildings as universal covers of certain CAT 0 complexes. Fundamental groups of these complexes will be used for expander constructions and for generating an infinite families of tessellated compact hyperbolic surfaces with large isometry groups.

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As a consequence the Swendsen-Wang algorithm for the ferromagnetic Ising model at any temperature has the same polynomial mixing time bound. A family of sets is said to be intersecting if any two sets in the family have nonempty intersection. Families of sets subject to various intersection conditions have been studied over the last fifty years and a common feature of many of the results in the area is that the extremal families are often quite asymmetric.

In this talk, I shall prove this conjecture. A Hamilton cycle in a directed graph G is a cycle that passes through every vertex of G. A Hamiltonian decomposition of G is a partition of its edge set into disjoint Hamilton cycles. In the late 60s Kelly conjectured that every regular tournament has a Hamilton decomposition. In this talk I will address the natural question of estimating the number of such decompositions of G. As a by product, we also obtain a new and much simpler proof for the approximate version of Kelly's conjecture.

We will consider a deceptively simple question formulated by Po-Shen Loh concerning sequences of integer triples.

## NSF Award Search: Award# - Studies in Algebraic and Enumerative Combinatorics

We shall discuss some recent developments following joint work with Tim Gowers, and mention a collection of generalisations and open problems. This is a work-in-progress talk. I will introduce a class of random geometric graphs where each node is associated to a real scalar or vectorial value, and the linking probability depends on a generalised visibility criterion. We will discuss a few basic properties of such graphs in the specific case of uniform point distribution with uniformly sampled node values, and we will provide a review of several interesting open problems and potential applications.

A d-dimensional bar-and-joint framework is said to be globally rigid if every d-dimensional framework with the same underlying graph and with the same edge lengths is congruent to it.

## Horizons of Combinatorics

In dimensions 1 and 2 global rigidity for generic frameworks can be characterised purely by properties of the underlying graph. In this talk I will describe these characterisations and discuss extensions to frameworks in 3-dimensions whose vertices are restricted to move on a fixed surface.

- When Wallflowers Dance: Becoming a Woman of Righteous Confidence;
- Combinatorics | Boundless Algebra;
- In the same section!

In particular when the surface is a generic family of concentric cylinders I will give a complete description of the graphs whose frameworks are generically globally rigid. We consider a degree proportional percolation model in which an infection spreads through a graph by infecting any vertex which has a fixed proportion of its neighbours already infected. This talk will present the background and solution to the problem of identifying the size of a minimal initial target set which will spread the infection to the entire graph.

S is a set of vertices of the discrete n- dimensional cube, the 'edge boundary' of S is the set of edges of the cube which join a vertex in S to a vertex outside S. There are several 'stability results' describing the structure of subsets of the discrete cube with 'small' edge-boundary; some, such as Friedgut's Junta theorem, have found many applications.

### Combinatorics

We conjecture that O m could be replaced by 2m. If time permits, generalizations to higher homology groups will be discussed. Joint work with B. DeMarco and J. Suppose H is a fixed graph. H-colourings of a graph G a. More than 15 years ago, Dyer and Greenhill considered the computational complexity of counting H-colourings, and demonstrated a dichotomy, in terms of the graph H, between polynomial time and P-complete.

That result was for exact counting, and, even now, there is only a partial complexity classification for approximate counting. It turns out that some interesting hereditary graph classes come into play in describing and proving the trichotomy result. We define a random geometric graph by choosing n points in a square and joining pairs of points if they are close to each other. It is natural to ask when standard graph properties such as connectedness, Hamiltonicity, et cetera typically occur in this model.

We consider the property of containing the square of a Hamilton cycle. Our main result is that for a typical point set the graph contains the square of a Hamilton cycle exactly when a simple local condition is satisfied at every vertex. Perhaps surprisingly, this property exhibits quite different behaviour in the binomial random graph. Furthermore, unlike in the case of connectedness and Hamiltonicity, the local condition is not simply a minimum degree condition.

The fundamental problem of distance geometry asks to find a realization of a finite, but partially specified, metric space in a Euclidean space of given dimension. Unless some structure is known about the structure of the instance, it is notoriously difficult to solve these problems computationally, and most methods will either not scale up to useful sizes, or will be unlikely to identify good solutions.

We propose a new heuristic algorithm based on a semidefinite programming formulation, a diagonally-dominant inner approximation of Ahmadi and Hall's, a randomized-type rank reduction method of Barvinok's, and a call to a local nonlinear programming solver. Short Bio: Leo Liberti obtained his Ph.