Past Combinatorial Theory Seminar

The homological Hall lemma is a topological tool that has recently been used to derive Hall type theorems for systems of disjoint representatives in hypergraphs.

After outlining the general method, we.ll describe one such theorem in some detail. The main ingredients in the proof are:

1) A relation between the spectral gap of a graph and the topological connectivity of its flag complex.

2) A new graph domination parameter defined via certain vector representations of the graph.

Joint work with R. Aharoni and E. Berger

How can one guarantee the presence of an oriented four-cycle in an oriented graph G? We shall see, that one way in which this can be done, is to demand that G contains no large `biased. subgraphs; where a `biased. subgraph simply means a subgraph whose orientation exhibits a strong bias in one direction.

Furthermore, we discuss the concept of biased subgraphs from another standpoint, asking: how can an oriented graph best avoid containing large biased subgraphs? Do random oriented graphs give the best examples? The talk is partially based on joint work with Omid Amini and Florian Huc.

Linear operators preserving non-vanishing properties are an important

tool in e.g. combinatorics, the Lee-Yang program on phase transitions, complex analysis, matrix theory. We characterize all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing when the variables are in prescribed open circular domains, which solves the higher dimensional counterpart of a long-standing classification problem going back to P\'olya-Schur. This also leads to a self-contained theory of multivariate stable polynomials and a natural framework for dealing with Lee-Yang and Heilmann-Lieb type problems in a uniform manner. The talk is based on joint work with Petter Brändén.

For any simple graph G and any positive integer lambda, let

P(G,lambda) denote the number of mappings f from V(G) to

{1,2,..,lambda} such that f(u) not= f(v) for every two adjacent

vertices u and v in G. It can be shown that

P(G,lambda) = \sum_{A \subseteq E} (-1)^{|A|} lambda^{c(A)}

where E is the edge set of G and c(A) is the number of components

of the spanning subgraph of G with edge set A. Hence P(G,lambda)

is really a polynomial of lambda. Many results on the chromatic

polynomial of a graph have been discovered since it was introduced

by Birkhoff in 1912. However, there are still many unsolved

problems and this talk will introduce the progress of some

problems and also some new problems proposed recently.

We call a family of permutations A in Sn 'intersecting' if any two permutations in A agree in at least one position. Deza and Frankl observed that an intersecting family of permutations has size at most (n-1)!; Cameron and Ku proved that equality is attained only by families of the form {σ in Sn: σ(i)=j} for i, j in [n].

We will sketch a proof of the following `stability' result: an intersecting family of permutations which has size at least (1-1/e + o(1))(n-1)! must be contained in {σ in Sn: σ(i)=j} for some i,j in [n]. This proves a conjecture of Cameron and Ku.

In order to tackle this we first use some representation theory and an eigenvalue argument to prove a conjecture of Leader concerning cross-intersecting families of permutations: if n >= 4 and A,B is a pair of cross-intersecting families in Sn, then |A||B|

The phase transition is a phenomenon that appears in natural sciences in various contexts. In the random graph theory, the phase transition refers to a dramatic change in the number of vertices in the largest components by addition of a few edges around a critical value, which was first discussed on the standard random graphs in the seminal paper by Erdos and Renyi. Since then, the phase transition has been a central theme of the random graph theory. In this talk we discuss the phase transition in random graphs with a given degree sequence and random graph processes with degree constraints.

A tree-rooted map is a planar map together with a

distinguished spanning tree. In the sixties, Mullin proved that the

number of tree-rooted maps with $n$ edges is the product $C_n C_{n+1}$

of two consecutive Catalan numbers. We will present a bijection

between tree-rooted maps (of size $n$) and pairs made of two trees (of

size $n$ and $n+1$ respectively) explaining this result.

Then, we will show that our bijection generalizes a correspondence by

Schaeffer between quandrangulations and so-called \emph{well labelled

trees}. We will also explain how this bijection can be used in order

to count bijectively several classes of planar maps

Let d be a fixed natural number. There is a theorem, due to Chvátal, Rodl,

Szemerédi and Trotter (CRST), saying that the Ramsey number of any graph G

with maximum degree d and n vertices is at most c(d)n, that is it grows

linearly with the size of n. The original proof of this theorem uses the

regularity lemma and the resulting dependence of c on d is of tower-type.

This bound has been improved over the years to the stage where we are now

grappling with proving the correct dependency, believed to be an

exponential in d. Our first main result is a proof that this is indeed the

case if we assume additionally that G is bipartite, that is, for a

bipartite graph G with n vertices and maximum degree d, we have r(G)

The partition function of the random cluster model on a graph $G$ is also known as its Potts model partition function. (Only the points at which it is evaluated differ in the two models.) This is a multivariate generalization of the Tutte polynomial of $G$, and encodes a wealth of enumerative information about spanning trees and forests, connected spanning subgraphs, electrical properties, and so on.

An elementary property of electrical networks translates into the statement that any two distinct edges are negatively correlated if one picks a spanning tree uniformly at random. Grimmett and Winkler have conjectured the analogous correlation inequalities for random forests or random connected spanning subgraphs. I'll survey some recent related work, partial results, and more specific conjectures, without going into all the gory details.

In the past decades the $G_{n,p}$ model of random graphs has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in $G_{n,p}$ appear independently.

The independence of the edges allows, for example, to obtain extremely tight bounds on the number of edges of $G_{n,p}$ and its degree sequence by straightforward applications of Chernoff bounds. This situation changes dramatically if one considers graph classes with structural side constraints. In this talk we show how recent progress in the construction of so-called Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer can be used to reduce the study of degree sequences and subgraph counts to properties of sequences of independent and identically distributed random variables -- to which we can then again apply Chernoff bounds to obtain extremely tight results. As proof of concept we study properties of random graphs that are drawn uniformly at random from the class consisting of the dissections of large convex polygons. We obtain very sharp concentration results for the number of vertices of any given degree, and for the number of induced copies of a given fixed graph.

A number of phylogenetic algorithms proceed by searching the space of all possible phylogenetic (leaf labeled) trees on a given set of taxa, using topological rearrangements and some optimality criterion. Recently, such an approach, called BSPR, has been applied to the balanced minimum evolution principle. Several computer studies have demonstrated the accuracy of BSPR in reconstructing the correct tree. It has been conjectured that BSPR is consistent, that is, when applied to an input distance that is a tree-metric, it will always converge to the (unique) tree corresponding to that metric. Here we prove that this is the case. Moreover, we show that even if the input distance matrix contains small errors relative to the tree-metric, then the BSPR algorithm will still return the corresponding tree.

Let d be a fixed natural number. There is a theorem, due to Chvátal, Rodl,

Szemerédi and Trotter (CRST), saying that the Ramsey number of any graph G

with maximum degree d and n vertices is at most c(d)n, that is it grows

linearly with the size of n. The original proof of this theorem uses the

regularity lemma and the resulting dependence of c on d is of tower-type.

This bound has been improved over the years to the stage where we are now

grappling with proving the correct dependency, believed to be an

exponential in d. Our first main result is a proof that this is indeed the

case if we assume additionally that G is bipartite, that is, for a

bipartite graph G with n vertices and maximum degree d, we have r(G)

Abstract: The Maximum Induced Planar Subgraph problem asks

for the largest set of vertices in a given input graph G

that induces a planar subgraph of G. Equivalently, we may

ask for the smallest set of vertices in G whose removal

leaves behind a planar subgraph. This problem has been

linked by Edwards and Farr to the problem of _fragmentability_

of graphs, where we seek the smallest proportion of vertices

in a graph whose removal breaks the graph into small (bounded

size) pieces. This talk describes some algorithms

developed for this problem, together with theoretical and

experimental results on their performance. The material

presented is joint work either with Keith Edwards (Dundee)

or Kerri Morgan (Monash).

Packings and coverings in graphs are related to two main problems of

graph theory, respectively error correcting codes and domination.

Given a set of words, an error correcting code is a subset such that

any two words in the subset are rather far apart, and can be

identified even if some errors occured during transmission. Error

correcting codes have been well studied already, and a famous example

of perfect error correcting codes are Hamming codes.

Domination is also a very old problem, initiated by some Chess problem

in the 1860's, yet Berge proposed the corresponding problem on graphs

only in the 1960's. In a graph, a subset of vertices dominates all the

graph if every vertex of the graph is neighbour of a vertex of the

subset. The domination number of a graph is the minimum number of

vertices in a dominating set. Many variants of domination have been

proposed since, leading to a very large literature.

During this talk, we will see how these two problems are related and

get into few results on these topics.

Phylogenetics is the reconstruction and analysis of 'evolutionary'

trees and graphs in biology (and related areas of classification, such as linguistics). Discrete mathematics plays an important role in the underlying theory. We will describe some of the ways in which concepts from combinatorics (e.g. poset theory, greedoids, cyclic permutations, Menger's theorem, closure operators, chordal graphs) play a central role. As well as providing an overview, we also describe some recent and new results, and outline some open problems.

We give different criteria for transience of branching Markov chains. These conditions enable us to give a classification of branching random walks in random environment (BRWRE) on Cayley graphs in recurrence and transience. This classification is stated explicitly for BRWRE on $\Z^d.$ Furthermore, we emphasize the interplay between branching Markov chains, the spectral radius, and some generating functions.

Hypergraph Transversals have been studied in Mathematics for a long time (e.g. by Berge) . Generating minimal transversals of a hypergraph is an important problem which has many applications in Computer Science, especially in database Theory, Logic, and AI. We give a survey of various applications and review some recent results on the complexity of computing all minimal transversals of a given hypergraph.

Glauber dynamics on $\mathbb{Z}^d$ is a dynamic representation of the zero-temperature Ising model, in which the spin (either $+$ or $-$) of each vertex updates, at random times, to the state of the majority of its neighbours. It has long been conjectured that the critical probability $p_c(\mathbb{Z}^d)$ for fixation (every vertex eventually in the same state) is $1/2$, but it was only recently proved (by Fontes, Schonmann and Sidoravicius) that $p_c(\mathbb{Z}^d)

The transportation problem (TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [Hitchcock, 1941] and independently by Koopmans in 1947 [Koopmans, 1948], and appears in any standard introductory course on operations research.

The mxn TP has m supply points and n demand points. Each supply Point i holds a quantity r_i, and each demand point j wants a quantity c_j, with the sum of femands equal to the sum of supplies. A solution to the problem can be written as a mxn matrix X with entries decision x_{ij} having value equal to the amount transported from supply point i to demand point j. The objective is to minimize total transportation costs when unit transporation costs between each supply and each demand point are given.

The set of feasible solutions of TP, is called the transportation polytope.

The 1-skeleton (edge graph) of this polytope is defined as the graph with vertices the vertices of the polytope and edges its 1-dimensional faces.

In 1957 W.M. Hirsch stated his famous conjecture cf. [Dantzig, 1963]) saying that any d-dimensional polytope with n facets has diameter at most n-d. So far the best bound for any polytope is O(n^{\log d+1}) [Kalai and Kleitman, 1992]. Any strongly polynomial bound is still lacking. Such bounds have been proved for some special classes of polytopes (for examples, see [Schrijver, 1995]). Among those are some special classes of transportation polytopes [Balinski, 1974],[Bolker, 1972] and the polytope of the dual of TP [Balinski, 1974].

The first strongly polynomial bound on the diameter of the transportation polytope was given by Dyer and Frieze [DyerFrieze, 1994]. Actually, they prove a bound on the diameter of any polytope {x|Ax=b} where A is a totally unimodular matrix. The proof is complicated and indirect, using the probabilistic method. Moreover, the bound is huge O(m^{16}n^3ln(mn))3) assuming m less than or equal to n.

We will give a simple proof that the diameter of the transportation polytope is less than 8(m+n-2). The proof is constructive: it gives an algorithm that describes how to go from any vertex to any other vertex on the transportation polytope in less than 8(m+n-2) steps along the edges.

According to the Hirsch Conjecture the bound on the TP polytope should be

m+n-1. Thus we are within a multiplicative factor 8 of the Hirsch bound.

Recently C. Hurkens refined our analysis and diminished the bound by a factor 2, arriving at 4(m+n-2). I will indicate the way he achieved this as well.

A graph property is a first order property if it can be written as a logic sentence with variables ranging over the vertices of the graph.

A sequence of random graphs (G_n)_n satisfies the zero-one law if the probability that G_n satisfies P tends to either zero or one for every first order property P. This is for instance the case for G(n,p) if p is fixed. I will survey some of the most important results on the G(n,p)-model and then proceed to discuss some work in progress on other graph models.

One of the main tools in studying sparse random graphs with independence between different edges is local comparison with branching processes. Recently, this method has been used to determine the asymptotic behaviour of the diameter (largest graph distance between two points that are in the same component) of various sparse random graph models, giving results for $G(n,c/n)$ as special cases. Nick Wormald and I have applied this method to $G(n,c/n)$ itself, obtaining a much stronger result, with a best-possible error term. We also obtain results as $c$ varies with $n$, including results almost all the way down to the phase transition.