Past Combinatorial Theory Seminar

We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. (Such a graph is said to be $r$-locally-$F$.) This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where $F$ is $\mathbb{L}^d$, the $d$-dimensional integer lattice. We obtain a characterisation of all the finite graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius $3$ in $\mathbb{L}^d$, for each integer $d$. These graphs have a very rigidly proscribed global structure, much more so than that of $(2d)$-regular graphs. (They can be viewed as quotient lattices in certain 'flat orbifolds'.) Our results are best possible in the sense that '3' cannot be replaced with '2'. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory. We will also discuss some results and open problems on the properties of a random n-vertex graph which is $r$-locally-$F$. This is all joint work with Itai Benjamini (Weizmann Institute of Science). 

We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. The sequence of maps has an almost sure limit G; we show that G is the distributional local limit of large, uniformly random 3-connected graphs.
A classical result of Brooks, Smith, Stone and Tutte associates squarings of rectangles to edge-rooted planar graphs. Our map growth procedure induces a growing sequence of squarings, which we show has an almost sure limit: an infinite squaring of a finite rectangle, which almost surely has a unique point of accumulation. We know almost nothing about the limit, but it should be in some way related to "Liouville quantum gravity".
Parts joint with Nicholas Leavitt.

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.


In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges. Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.


Based on joint work with Oliver Riordan.

A system of linear equations is called partition regular if, whenever the naturals are finitely coloured, there is a monochromatic solution of the equations. Many of the classical theorems of Ramsey Theory may be phrased as assertions that certain systems are partition regular.

What happens if we are colouring not the naturals but the (non-zero) integers, rationals, or reals instead? After some background, we will give various recent results.

Erdos, Faudree, Gould, Gyarfas, Rousseau and Schelp, conjectured that
whenever the edges of a complete graph are coloured using three colours
there always exists a set of at most three vertices which have at least
two-thirds of their neighbours in one of the colours.  We will describe a
proof of this conjecture. This is joint work with Rahil Baber

Fix positive integers p and q with 2 \leq q \leq {p \choose 2}. An edge-colouring of the complete graph K_n is said to be a (p, q)-colouring if every K_p receives at least q different colours. The function f(n, p, q) is the minimum number of colours that are needed for K_n to have a (p,q)-colouring. This function was introduced by Erdos and Shelah about 40 years ago, but Erdos and Gyarfas were the first to study the function in a systematic way. They proved that f(n, p, p) is polynomial in n and asked to determine the maximum q, depending on p, for which f(n,p,q) is subpolynomial in n. We prove that the answer is p-1.

We also discuss some related questions.

Joint work with Jacob Fox, Choongbum Lee and Benny Sudakov.

I will discuss a novel approach to estimating communication costs of an algorithm (also known as its I/O complexity), which is based on small-set expansion for computational graphs. Various applications and implications will be discussed as well, mostly having to do with linear algebra algorithms. This includes, in particular, first known (and tight) bounds on communication complexity of fast matrix multiplication.

Joint work with Grey Ballard, James Demmel, Benjamin Lipshitz and Oded Schwartz.

We discuss some questions related to coloring the edge/vertices of randomgraphs. In particular we look at
(i) The game chromatic number;
(ii) Rainbow Matchings and Hamilton cycles;
(iii) Rainbow Connection;
(iv) Zebraic Colorings.

Several objects can be associated to a list of vectors with integer coordinates: among others, a family of tori called toric arrangement, a convex polytope called zonotope, a function called vector partition function; these objects have been described in a recent book by De Concini and Procesi. The linear algebra of the list of vectors is axiomatized by the combinatorial notion of a matroid; but several properties of the objects above depend also on the arithmetics of the list. This can be encoded by the notion of a "matroid over Z". Similarly, applications to tropical geometry suggest the introduction of matroids over a discrete valuation ring.Motivated by the examples above, we introduce the more general notion of a "matroid over a commutative ring R". Such a matroid arises for example from a list of elements in a R-module. When R is a Dedekind domain, we can extend the usual properties and operations holding for matroids (e.g., duality). We can also compute the Tutte-Grothendieck ring of matroids over R; the class of a matroid in such a ring specializes to several invariants, such as the Tutte polynomial and the Tutte quasipolynomial. We will also outline other possible applications and open problems. (Joint work with Alex Fink).

We give a new proof of the Frankl-Rödl theorem on set systems with a forbidden intersection. Our method extends to codes with forbidden distances, where over large alphabets our bound is significantly better than that obtained by Frankl and Rödl. One consequence of our result is a Frankl-Rödl type theorem for permutations with a forbidden distance. Joint work with Peter Keevash.

A Steiner Triple System on a set X is a collection T of 3-element subsets of X such that every pair of elements of X is contained in exactly one of the triples in T. An example considered by Plücker in 1835 is the affine plane of order three, which consists of 12 triples on a set of 9 points. Plücker observed that a necessary condition for the existence of a Steiner Triple System on a set with n elements is that n be congruent to 1 or 3 mod 6. In 1846, Kirkman showed that this necessary condition is also sufficient.

In 1853, Steiner posed the natural generalisation of the question: given integers q and r, for which n is it possible to choose a collection Q of q-element subsets of an n-element set X such that any r elements of X are contained in exactly one of the sets in Q? There are some natural necessary divisibility conditions generalising the necessary conditions for Steiner Triple Systems. The Existence Conjecture states that for all but finitely many n these divisibility conditions are also sufficient for the existence of general Steiner systems (and more generally designs).

We prove the Existence Conjecture, and more generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition.

We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees.

Joint work with Christian Borgs, Jennifer T. Chayes, and Henry Cohn.

We consider the parameter $a(H)$, which is the smallest a such that if $|E(G)|$ is at least/exceeds $a|V(H)|/2$ then $G$ has an $H$-minor. We are especially interested in sparse $H$ and in bounding $a(H)$ as a function of $|E(H)|$ and $|V(H)|$. This is joint work with David Wood.

Nesetril and Ossona de Mendez introduced a new notion of convergence of graphs called FO convergence. This notion can be viewed as a unified notion of convergence of dense and sparse graphs. In particular, every FO convergent sequence of graphs is convergent in the sense of left convergence of dense graphs as studied by Borgs, Chayes, Lovasz, Sos, Szegedy, Vesztergombi and others, and every FO convergent sequence of graphs with bounded maximum degree is convergent in the Benjamini-Schramm sense.

FO convergent sequences of graphs can be associated with a limit object called modeling. Nesetril and Ossona de Mendez showed that every FO convergent sequence of trees with bounded depth has a modeling. We extend this result

to all FO convergent sequences of trees and discuss possibilities for further extensions.

The talk is based on a joint work with Martin Kupec and Vojtech Tuma.

We prove a generalization of Helly's theorem concerning intersections of convex sets that has an interesting voting theory interpretation. We then
consider various extensions in which compelling mathematical problems are motivated from very natural questions in the voting context.

The Ramsey number $R(K_s, Q_n)$ is the smallest integer $N$ such that every red-blue colouring of the edges of the complete graph $K_N$ contains either a red $n$-dimensional hypercube, or a blue clique on $s$ vertices. Note that $N=(s-1)(2^n -1)$ is not large enough, since we may colour in red $(s-1)$ disjoint cliques of cardinality $2^N -1$ and colour the remaining edges blue. In 1983, Burr and Erdos conjectured that this example was the best possible, i.e., that $R(K_s, Q_n) = (s-1)(2^n -1) + 1$ for every positive integer $s$ and sufficiently large $n$. In a recent breakthrough, Conlon, Fox, Lee and Sudakov proved the conjecture up to a multiplicative constant for each $s$. In this talk we shall sketch the proof of the conjecture and discuss some related problems.

(Based on joint work with Gonzalo Fiz Pontiveros, Robert Morris, David Saxton and Jozef Skokan)

The Tutte polynomial of a graph $G$ is a two-variable polynomial $T(G;x,y)$, which encodes much information about~$G$. The number of spanning trees in~$G$, the number of acyclic orientations of~$G$, and the partition function of the $q$-state Potts model are all specialisations of the Tutte polynomial. Jackson and Sokal have studied the sign of the Tutte polynomial, and identified regions in the $(x,y)$-plane where it is ``essentially determined'', in the sense that the sign is a function of very simple characteristics of $G$, e.g., the number of vertices and connected components of~$G$. It is natural to ask whether the sign of the Tutte polynomial is hard to compute outside of the regions where it is essentially determined. We show that the answer to this question is often an emphatic ``yes'': specifically, that determining the sign is \#P-hard. In such cases, approximating the Tutte polynomial with small relative error is also \#P-hard, since in particular the sign must be determined. In the other direction, we can ask whether the Tutte polynomial is easy to approximate in regions where the sign is essentially determined. The answer is not straightforward, but there is evidence that it often ``no''. This is joint work with Leslie Ann Goldberg (Oxford).

Perfect matchings are fundamental objects of study in graph theory. There is a substantial classical theory, which cannot be directly generalised to hypergraphs unless P=NP, as it is NP-complete to determine whether a hypergraph has a perfect matching. On the other hand, the generalisation to hypergraphs is well-motivated, as many important problems can be recast in this framework, such as Ryser's conjecture on transversals in latin squares and the Erdos-Hanani conjecture on the existence of designs. We will discuss a characterisation of the perfect matching problem for uniform hypergraphs that satisfy certain density conditions (joint work with Richard Mycroft), and a polynomial time algorithm for determining whether such hypergraphs have a perfect matching (joint work with Fiachra Knox and Richard Mycroft).

Let $G$ be an addable minor-closed class of graphs. We prove that a zero-one law holds in monadic second-order logic (MSO) for connected graphs in G, and a convergence law in MSO for all graphs in $G$. For each surface $S$, we prove the existence of a zero-one law in first order logic (FO) for connected graphs embeddable in $S$, and a convergence law in FO for all graphs embeddable in $S$. Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface $S$. If $G$ is an addable minor-closed class, we prove that the closure of the set of limiting probabilities is a finite union of intervals, and it is the same for FO and MSO. For the class of planar graphs it consists of exactly 108 intervals. We give examples of non-addable classes where the results are quite different: for instance, the zero-one law does not hold for caterpillars, even in FO. This is joint work with Peter Heinig, Tobias Müller and Anusch Taraz.

An independent set in an $r$-uniform hypergraph is a subset of the vertices that contains no edges. A container for the independent set is a superset of it. It turns out to be desirable for many applications to find a small collection of containers, none of which is large, but which between them contain every independent set. ("Large" and "small" have reasonable meanings which will be explained.)

Applications include giving bounds on the list chromatic number of hypergraphs (including improving known bounds for graphs), counting the solutions to equations in Abelian groups, counting Sidon sets, establishing extremal properties of random graphs, etc.

The work is joint with David Saxton.

The first application of Szemeredi's regularity method was the following celebrated Ramsey-Turan result proved by Szemeredi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1))N^2 edges. Four years later, Bollobas and Erdos gave a surprising geometric construction, utilizing the isodiametric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollobas and Erdos in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8.

These problems have received considerable attention and remained one of the main open problems in this area.  More generally, it remains an important problem to determine if, for certain applications of the regularity method, alternative proofs exist which avoid using the regularity lemma and give better quantitative estimates.  In this work, we develop new regularity-free methods which give nearly best-possible bounds, solving the various open problems concerning this critical window.

Joint work with Jacob Fox and Yufei Zhao.

Consider the complete graph on n vertices with independent and identically distributed edge-weights having some absolutely continuous distribution. The minimum spanning tree (MST) is simply the spanning subtree of smallest weight. It is straightforward to construct the MST using one of several natural algorithms. Kruskal's algorithm builds the tree edge by edge starting from the globally lowest-weight edge and then adding other edges one by one in increasing order of weight, as long as they do not create any cycles. At each step of this process, the algorithm has generated a forest, which becomes connected on the final step. In this talk, I will explain how it is possible to exploit a connection between the forest generated by Kruskal's algorithm and the Erd\"os-R\'enyi random graph in order to prove that $M_n$, the MST of the complete graph, possesses a scaling limit as $n$ tends to infinity. In particular, if we think of $M_n$ as a metric space (using the graph distance), rescale edge-lengths by $n^{-1/3}$, and endow the vertices with the uniform measure, then $M_n$ converges in distribution in the sense of the Gromov-Hausdorff-Prokhorov distance to a certain random measured real tree.

This is joint work with Louigi Addario-Berry (McGill), Nicolas Broutin (INRIA Paris-Rocquencourt) and Grégory Miermont (ENS Lyon).

The ``k-commodity flow problem'' is: we are given k pairs of vertices of a graph, and we ask whether there are k flows in the graph, where the ith flow is between the ith pair of vertices, and has total value one, and for each edge, the sum of the absolute values of the flows along it is at most one. We may also require the flows to be 1/2-integral, or indeed 1/p-integral for some fixed p.

If the problem is feasible (that is, the desired flows exist) then it is still feasible after contracting any edge, so let us say a flow problem is ``critical'' if it is infeasible, but becomes feasible when we contract any edge. In many special cases, all critical instances have only two vertices, but if we ask for integral flows (that is, p = 1, essentially the edge-disjoint paths problem), then there arbitrarily large critical instances, even with k = 2. But it turns out that p = 1 is the only bad case; if p>1 then all critical instances have bounded size (depending on k, but independent of p), and the same is true if there is no integrality requirement at all.

The proof gives rise to a very simple algorithm for the k edge-disjoint paths problem in 4-edge-connected graphs.