Combinatorial Theory Seminar (past)

Tue, 14/05 14:30	Maria Chudnovsky (Columbia)	Combinatorial Theory Seminar	L3
	3-coloring graphs with no induced 6-edge paths
	Since graph-coloring is an NP-complete problem in general, it is natural to ask how the complexity changes if the input graph is known not to contain a certain induced subgraph H. Due to results of Kaminski and Lozin, and Hoyler, the problem remains NP-complete, unless H is the disjoint union of paths. Recently the question of coloring graphs with a fixed-length induced path forbidden has received considerable attention, and only a few cases of that problem remain open for k-coloring when k>=4. However, little is known for 3-coloring. Recently we have settled the first open case for 3-coloring; namely we showed that 3-coloring graphs with no induced 6-edge paths can be done in polynomial time. In this talk we will discuss some of the ideas of the algorithm. This is joint work with Peter Maceli and Mingxian Zhong.

Tue, 07/05 14:30	Joel Ouaknine (University of Oxford)	Combinatorial Theory Seminar	L3
	Positivity problems for low-order linear recurrence sequences
	We consider two decision problems for linear recurrence sequences(LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, and for simple LRS (i.e. whose characteristic polynomial has no repeated roots) of order 9 or less. Moreover, we show by way of hardness that extending the decidability of either problem to LRS of order 6 would entail major breakthroughs in analytic number theory, more precisely in the field of Diophantine approximation of transcendental numbers.This talk is based on a recent paper, available athttp://www.cs.ox.ac.uk/people/joel.ouaknine/publications/positivity13abs.html joint with James Worrell and Matt Daws.

Tue, 23/04 14:30	Robert Leese (Smith Institute)	Combinatorial Theory Seminar	L3
	Inside the 4G Spectrum Auction
	The recently completed auction for 4G mobile spectrum was the most importantcombinatorial auction ever held in the UK. In general, combinatorial auctions allow bidders to place individual bids on packages of items,instead of separate bids on individual items, and this feature has theoretical advantages for bidders and sellers alike. The accompanying challenges of implementation have been the subject of intense work over the last few years, with the result that the advantages of combinatorial auctions can now be realised in practice on a large scale. Nowhere has this work been more prominent than in auctions for radio spectrum. The UK's 4G auction is the most recent of these and the publication by Ofcom (the UK's telecommunications regulator) of the auction's full bidding activity creates a valuable case study of combinatorial auctions in action.

Tue, 05/03 14:30	Dan Hefetz (Birmingham)	Combinatorial Theory Seminar	L3
	Optimal covers of random graphs with Hamilton cycles
	We prove that if $\frac{\log^{117} n}{n} \leq p \leq 1 - n^{-1/8}$ , then asymptotically almost surely the edges of $G(n,p)$ can be covered by $\lceil \Delta(G(n,p))/2 \rceil$ Hamilton cycles. This is clearly best possible and improves an approximate result of Glebov, Krivelevich and Szabó, which holds for $p \geq n^{-1 + \varepsilon}$ . Based on joint work with Daniela Kuhn, John Lapinskas and Deryk Osthus.

Tue, 26/02 14:30	Oleg Pikhurko (Warwick)	Combinatorial Theory Seminar	L3
	Limit method in extremal combinatorics
	Razborov's flag algebras provide a formal system for operating with asymptotic inequalities between subgraph densities, allowing to do extensive "book-keeping" by a computer. This novel use of computers led to progress on many old problems of extremal combinatorics. In some cases, finer structural information can be derived from a flag algebra proof by by using the Removal Lemma or graph limits. This talk will overview this approach.

Tue, 19/02 14:30	Karen Johannson (Bristol)	Combinatorial Theory Seminar	L3
	Bootstrap percolation on infinite trees
	While usual percolation concerns the study of the connected components of random subgraphs of an infinite graph, bootstrap percolation is a type of cellular automaton, acting on the vertices of a graph which are in one of two states: `healthy' or `infected'. For any positive integer $r$ , the $r$ -neighbour bootstrap process is the following update rule for the states of vertices: infected vertices remain infected forever and each healthy vertex with at least $r$ infected neighbours becomes itself infected. These updates occur simultaneously and are repeated at discrete time intervals. Percolation is said to occur if all vertices are eventually infected. As it is often difficult to determine precisely which configurations of initially infected vertices percolate, one often considers a random case, with each vertex infected independently with a fixed probability $p$ . For an infinite graph, of interest are the values of $p$ for which the probability of percolation is positive. I will give some of the history of this problem for regular trees and present some new results for bootstrap percolation on certain classes of randomly generated trees: Galton–Watson trees.

Tue, 12/02 14:30	Veselin Jungic (Simon Fraser University)	Combinatorial Theory Seminar	L3
	From monotone arithmetic progressions to bounded additive complexity in infinite words
	I will describe how a search for the answer to an old question about the existence of monotone arithmetic progressions in permutations of positive integers led to the study of infinite words with bounded additive complexity. The additive complexity of a word on a finite subset of integers is defined as the function that, for a positive integer $n$ , counts the maximum number of factors of length $n$ , no two of which have the same sum.

Tue, 05/02 14:30	David Ellis (Queen Mary)	Combinatorial Theory Seminar	L3
	Juntas, stability and isoperimetric inequalities in the symmetric group
	Results of Bourgain and Kindler-Safra state that if $f$ is a Boolean function on $\{0,1\}^n$ , and the Fourier transform of $f$ is highly concentrated on low frequencies, then $f$ must be close to a ‘junta’ (a function depending upon a small number of coordinates). This phenomenon is known as ‘Fourier stability’, and has several interesting consequences in combinatorics, theoretical computer science and social choice theory. We will describe some of these, before turning to the analogous question for Boolean functions on the symmetric group. Here, genuine stability does not occur; it is replaced by a weaker phenomenon, which we call ‘quasi-stability’. We use our 'quasi-stability' result to prove an isoperimetric inequality for $S_n$ which is sharp for sets of size $(n-t)!$ , when $n$ is large. Several open questions remain. Joint work with Yuval Filmus (University of Toronto) and Ehud Friedgut (Weizmann Institute).

Tue, 29/01 14:30	Mireille Bousquet-Melou (University of Bordeaux)	Combinatorial Theory Seminar	L3
	Self-avoiding walks in a half-plane
	A self-avoiding walk on a lattice is a walk that never visits the same vertex twice. Self-avoiding walks (SAW) have attracted interest for decades, first in statistical physics, where they are considered as polymer models, and then in combinatorics and in probability theory (the first mathematical contributions are probably due to John Hammersley, from Oxford, in the early sixties). However, their properties remain poorly understood in low dimension, despite the existence of remarkable conjectures. About two years ago, Duminil-Copin and Smirnov proved an "old" and remarkable conjecture of Nienhuis (1982), according to which the number of SAWs of length n on the honeycomb (hexagonal) lattice grows like mu^n, with mu=sqrt(2 +sqrt(2)). This beautiful result has woken up the hope to prove other simple looking conjectures involving these objects. I will thus present the proof of a younger conjecture (1995) by Batchelor and Yung, which deals with SAWs confined to a half-plane and interacting with its boundary.(joint work with N. Beaton, J. de Gier, H. Duminil-Copin and A. Guttmann)

Tue, 22/01 14:30	Eoin Long (University of London)	Combinatorial Theory Seminar	L3
	Long paths and cycles in subgraphs of the cube
	Let $Q_n$ denote the graph of the $n$ -dimensional cube with vertex set $\{0, 1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. Suppose $G$ is a subgraph of $Q_n$ with average degree at least $d$ . How long a path can we guarantee to find in $G$ ? My aim in this talk is to show that $G$ must contain an exponentially long path. In fact, if $G$ has minimum degree at least $d$ then $G$ must contain a path of length $2^d − 1$ . Note that this bound is tight, as shown by a $d$ -dimensional subcube of $Q^n$ . I hope to give an overview of the proof of this result and to discuss some generalisations.

Tue, 27/11/2012
14:30

Annika Heckel (Oxford)

Combinatorial Theory Seminar

SR1

The hitting time of rainbow connectivity two

Rainbow connectivity is a new concept for measuring the connectivity of a graph which was introduced in 2008 by Chartrand, Johns, McKeon and Zhang. In a graph G with a given edge colouring, a rainbow path is a path all of whose edges have distinct colours. The minimum number of colours required to colour the edges of G so that every pair of vertices is joined by at least one rainbow path is called the rainbow connection number rc(G) of the graph G.

For any graph G, rc(G) >= diam(G). We will discuss rainbow connectivity in the random graph setting and present the result that for random graphs, rainbow connectivity 2 happens essentially at the same time as diameter 2. In fact, in the random graph process, with high probability the hitting times of diameter 2 and of rainbow connection number 2 coincide

Tue, 20/11/2012 14:30	Matthias Lenz (Merton College)	Combinatorial Theory Seminar	SR1
	"Interpolation, box splines, and lattice points in zonotopes"
	Given a finite list of vectors X in $\R^d$, one can define the box spline $B_X$. Box splines are piecewise polynomial functions that are used in approximation theory. They are also interesting from a combinatorial point of view and many of their properties solely depend on the structure of the matroid defined by the list X. The support of the box spline is a certain polytope called zonotope Z(X). We will show that if the list X is totally unimodular, any real-valued function defined on the set of lattice points in the interior of Z(X) can be extended to a function on Z(X) of the form $p(D)B_X$ in a unique way, where p(D) is a differential operator that is contained in the so-called internal P-space. This was conjectured by Olga Holtz and Amos Ron. The talk will focus on combinatorial aspects and all objects mentioned above will be defined. (arXiv:1211.1187)

Tue, 13/11/2012 14:30	Asaf Ferber (Tel Aviv)	Combinatorial Theory Seminar	SR1
	Counting and packing Hamilton cycles in dense graphs and oriented graphs
	In this talk we present a general method using permanent estimates in order to obtain results about counting and packing Hamilton cycles in dense graphs and oriented graphs. As a warm up we prove that every Dirac graph $G$ contains at least $(reg(G)/e)^n$ many distinct Hamilton cycles, where $reg(G)$ is the maximal degree of a spanning regular subgraph of $G$ . We continue with strengthening a result of Cuckler by proving that the number of oriented Hamilton cycles in an almost $cn$ -regular oriented graph is $(cn/e)^n(1+o(1))^n$ , provided that $c$ is greater than $3/8$ . Last, we prove that every graph $G$ of minimum degree at least $n/2+\epsilon n$ contains at least $reg_{even}(G)-\epsilon n$ edge-disjoint Hamilton cycles, where $reg_{even}(G)$ is the maximal even degree of a spanning regular subgraph of $G$ . This proves an approximate version of a conjecture made by Osthus and Kühn. Joint work with Michael Krivelevich and Benny Sudakov.

Tue, 30/10/2012 14:30	Oliver Riordan (Oxford)	Combinatorial Theory Seminar	SR1
	Local limit theorems for giant components
	In an Erdős–Rényi random graph above the phase transition, i.e., where there is a giant component, the size of (number of vertices in) this giant component is asymptotically normally distributed, in that its centred and scaled size converges to a normal distribution. This statement does not tell us much about the probability of the giant component having exactly a certain size. In joint work with Béla Bollobás we prove a `local limit theorem' answering this question for hypergraphs; the graph case was settled by Luczak and Łuczak. The proof is based on a `smoothing' technique, deducing the local limit result from the (much easier) `global' central limit theorem.

Tue, 23/10/2012 16:30	Charles Semple (University of Canterbury)	Combinatorial Theory Seminar	SR2
	Realising evolutionary trees with local information
	Results that say local information is enough to guarantee global information provide the theoretical underpinnings of many reconstruction algorithms in evolutionary biology. Such results include Buneman's Splits-Equivalence Theorem and the Tree-Metric Theorem. The first result says that, for a collection $\mathcal C$ of binary characters, pairwise compatibility is enough to guarantee compatibility for $\mathcal C$ , that is, there is a phylogenetic (evolutionary) tree that realises $\mathcal C$ . The second result says that, for a distance matrix $D$ , if every $4\times 4$ distance submatrix of $D$ is realisable by an edge-weighted phylogenetic tree, then $D$ itself is realisable by such a tree. In this talk, we investigate these and other results of this type. Furthermore, we explore the closely-related task of determining how much information is enough to reconstruct the correct phylogenetic tree.

Tue, 23/10/2012 14:30	Van Vu (Yale)	Combinatorial Theory Seminar	SR1
	Law of the determinant
	Consider random matrices with independent entries (in both hermitian and non-hermtian setting). An old and basic question is: What is the law of the determinant ? I am going to give a survey about this problem, focusing on recent developments and new techniques, along with several open questions.(partially based on joint works with H. Nguyen and T. Tao).

Tue, 09/10/2012 14:30	Sinai Robins (Nanyang Technological University)	Combinatorial Theory Seminar	L3
	Tiling Euclidean space with a polytope, by translations
	We study the problem of covering R^d by overlapping translates of a convex polytope, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov and Minkowski, and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile R^d. By contrast, for k ≥ 2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational, centrally symmetric polytope, and if P has centrally symmetric facets, then P must k-tile R^d for some positive integer k.

Tue, 22/05/2012 14:30	Jozef Skokan (LSE)	Combinatorial Theory Seminar	L3
	Strong Ramsey saturation for cycles
	We call a graph $H$ Ramsey-unsaturated if there is an edge in the complement of $H$ such that the Ramsey number $r(H)$ of $H$ does not change upon adding it to $H$ . This notion was introduced by Balister, Lehel and Schelp who also showed that cycles (except for $C_4$ ) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle $C_n$ , unless $n$ is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement: If a graph $H$ is obtained by adding a linear number of chords to a cycle $C_n$ , then $r(H)=r(C_n)$ , as long as the maximum degree of $H$ is bounded, $H$ is either bipartite (for even $n$ ) or almost bipartite (for odd $n$ ), and $n$ is large. This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method.

Tue, 08/05/2012 14:30	Hao Huang (UCLA)	Combinatorial Theory Seminar	L3
	Extremal Problems in Eulerian Digraphs
	Graphs and digraphs behave quite differently, and many classical results for graphs are often trivially false when extended to general digraphs. Therefore it is usually necessary to restrict to a smaller family of digraphs to obtain meaningful results. One such very natural family is Eulerian digraphs, in which the in-degree equals out-degree at every vertex. In this talk, we discuss several natural parameters for Eulerian digraphs and study their connections. In particular, we show that for any Eulerian digraph G with n vertices and m arcs, the minimum feedback arc set (the smallest set of arcs whose removal makes G acyclic) has size at least $m^2/2n^2+m/2n$ , and this bound is tight. Using this result, we show how to find subgraphs of high minimum degrees, and also long cycles in Eulerian digraphs. These results were motivated by a conjecture of Bollobás and Scott. Joint work with Ma, Shapira, Sudakov and Yuster

Tue, 24/04/2012 14:30	Choongbum Lee (UCLA)	Combinatorial Theory Seminar	L3
	Large and judicious bisections of graphs
	It is very well known that every graph on $n$ vertices and $m$ edges admits a bipartition of size at least $m/2$ . This bound can be improved to $m/2 + (n-1)/4$ for connected graphs, and $m/2 + n/6$ for graphs without isolated vertices, as proved by Edwards, and Erdös, Gyárfás, and Kohayakawa, respectively. A bisection of a graph is a bipartition in which the size of the two parts differ by at most 1. We prove that graphs with maximum degree $o(n)$ in fact admit a bisection which asymptotically achieves the above bounds.These results follow from a more general theorem, which can also be used to answer several questions and conjectures of Bollobás and Scott on judicious bisections of graphs.Joint work with Po-Shen Loh and Benny Sudakov