Jaroslav Nesetril

Duality and Polynomial Testing of Tree Homomorphisms

Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.

Chromatically optimal rigid graphs

The purpose of this paper is to prove that a given graph G with chromatic number k (k finite or infinite) is a subgraph of a k-chromatic rigid graph H if and only if G does not contain a complete graph with k vertices. This solves a problem of L. Babai and J. Nesetřil.

For graphs there are only four types of hereditary Ramsey classes

A Ramsey class is defined as a hereditary class K of structures which has the A-partition property for every A E K; see [4,6]. This notion generalizes the classical Ramsey theorem [S] which in this setting claims that the class of all finite complete graphs is Ramsey. Ramsey classes of graphs, relations, and set systems were studied in [l, 6, 71 where several basic classes were proved to be Ramsey. Here we combine these results with a result of Lachlan and Woodrow [3] and we obtain a complete list of Ramsey classes of graphs. As we shall see there are exactly four basic types of classes of graphs which are Ramsey. All other classes may be obtained by unions. It should be stressed that being a Ramsey class is a. very restrictive property and usually it is not easy to establish this fact. We define below a much weaker property (“pair-Ramsey class”); nevertheless we prove that Ramsey and pair-Ramsey properties coincide. This is quite surprising and there is no direct proof of this fact (our proof is indirect via [3]). The paper is organized as follows: Section 1 contains basic notions and properties of Ramsey classes of graphs; Section 2 reviews amalgams and amalgamation classes introduced in [3]; in Section 3 we define the notion of a pair-Ramsey class and derive basic properties. In Section 4 we prove the main result.

Colouring Relatives of Intervals on the Plane, II

Obviously, R ⊆ I ⊆ S and thus f (R, k) ≤ f (I, k) ≤ f (S, k) and g(R, k) ≤ g(I, k) ≤ g(S, k) for every k. Note also that f (G, 2) = g(G, 4) for every family G. The results in McGuiness [5] on coloring intersection graphs of arcwise connected sets imply that f (R, k) < ∞ for each integer k. No such facts are known for f (I, k) and f (S, k) and, in fact, the following problems motivated our research:

Representations of Graphs by Means of Products and Their Complexity

I. A graph G is a pair (V,E) where V is a finite set and E is either a set of ordered pairs from V (i.e. E ~V × V) or a set of unordered pairs from V (i.e. E ~ [V3 2 ) . In the former case G is called an oriented graph in the latter an unoriented graph . Elements of the set E are called edges. An edge e g E is indicated by [x,y] = e where the bracketing means consistently in the whole formula either (x,y) (an arrow an ordered edge) or ~x,y} Ca link an unordered edge) .

Strong Ramsey theorems for Steiner systems

It is shown that the class of partial Steiner (fc, Z)-systems has the edge Ramsey property, i.e., we prove that for every partial Steiner (k, i)-system Q there exists a partial Steiner (fc, Z)-system )i such that for every partition of the edges of H into two classes one can find an induced monochromatic copy of Q. As an application we get that the class of all graphs without cycles of lengths 3 and 4 has the edge Ramsey property. This solves a longstanding problem in the area.

More about subcolorings

A subcoloring is a vertex coloring of a graph in which every color class induces a disjoint union of cliques. We derive a number of results on the combinatorics, the algorithmics, and the complexity of subcolorings.On the negative side, we prove that 2-subcoloring is NP-hard for comparability graphs, and that 3-subcoloring is NP-hard for AT-free graphs and for complements of planar graphs. On the positive side, we derive polynomial time algorithms for 2-subcoloring of complements of planar graphs, and for r-subcoloring of interval and of permutation graphs. Moreover, we prove asymptotically best possible upper bounds on the subchromatic number of interval graphs, chordal graphs, and permutation graphs in terms of the number of vertices.

From sparse graphs to nowhere dense structures: decompositions, independence, dualities and limits

We discuss a formality result for 2-dimensional topological field theories which are based on a semi-simple Frobenius algebra: namely, when sufficiently constrained by structural axioms, the complete theory is determined by the Frobenius algebra and the grading information. The structural constraints apply to Gromov-Witten theory of a variety whose quantum cohomology is semi-simple. Some open questions about semi-simple field theories are mentioned in the final section.This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical which is a purely homotopy theoretical condition implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are topologically rigid. The article contains new results about product decompositions of closed aspherical manifolds and an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension greater or equal to six as boundary. At the end we describe (winking) our universe of closed manifolds.Wheeled props is one the latest species found in the world of operads and props. We attempt to give an elementary introduction into the main ideas of the theory of wheeled props for beginners, and also a survey of its most recent major applications (ranging from algebra and geometry to deformation theory and Batalin-Vilkovisky quantization) which might be of interest to experts.We discuss scaling limits of random planar maps chosen uniformly at random in a certain class. This leads to a universal limiting space called the Brownian map, which is viewed as a random compact metric space. The Brownian map can be obtained as a quotient of the continuous random tree called the CRT, for an equivalence relation which is defined in terms of Brownian labels assigned to the vertices of the CRT. We discuss the known properties of the Brownian map. In particular, we give a complete description of the geodesics starting from the distinguished point called the root. We also discuss applications to various properties of large random planar maps.We show how to extend the method used in [22] to prove uniqueness of solutions to a family of several nonlocal equations containing aggregation terms and aggregation/diusion competition. They contain several mathematical biology models proposed in macroscopic descriptions of swarming and chemotaxis for the evolution of mass densities of individuals or cells. Uniqueness is shown for bounded nonnegative mass-preserving weak solutions without diusion. In diusive cases, we use a coupling method [16, 33] and thus, we need an stochastic representation of the solution to hold. In summary, our results show, modulo certain technical hypotheses, that nonnegative mass-preserving solutions remain unique as long as their L 1 -norm is controlled in time.We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the algebraic complexity theory as well as the general error analysis for matrix multiplication and related problems. We emphasize the central role played by the matrix multiplication problem and discuss historical and modern approaches to its solution.The European Congress of Mathematics, held every four years, has established itself as a major international mathematical event. Following those in Paris (1992), Budapest (1996), Barcelona (2000) and Stockholm (2004), the Fifth European Congress of Mathematics (5ECM) took place in Amsterdam, The Netherlands, July 14-18, 2008, with about 1000 participants from 68 different countries. Ten plenary and thirty-three invited lectures were delivered. Three science lectures outlined applications of mathematics in other sciences: climate change, quantum information theory and population dynamics. As in the four preceding EMS congresses, ten EMS prizes were granted to very promising young mathematicians. In addition, the Felix Klein Prize was awarded, for the second time, for an application of mathematics to a concrete and difficult industrial problem. There were twenty-two minisymposia, spread over the whole mathematical area. Two round table meetings were organized: one on industrial mathematics and one on mathematics and developing countries. As part of the 44th Nederlands Mathematisch Congres, which was embedded in 5ECM, the so-called Brouwer lecture was presented. It is the Netherlands most prestigious award in mathematics, organized every three years by the Royal Dutch Mathematical Society. Information about Brouwer was given in an invited historical lecture during the congress. These proceedings contain a selection of the contributions to the congress.I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact play a crucial role in Delsartes method for finding bounds for the density of sphere packings on spheres and Euclidean spaces. One of the most excited applications of Delsartes method is a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed the lower. We have found an extension of the Delsarte method that allows to solve the kissing number problem (as well as the one-sided kissing number problem) in dimensions three and four. In this paper we also will discuss the maximal cardinalities of spherical two-distance sets. Using the so-called polynomial method and Delsartes method these cardinalities can be determined for all dimensions

Universal H-colorable graphs without a given configuration

n<40

On maximal finite antichains in the homomorphism order of directed graphs

. Recently, were found extensions of Schoenbergs theorem for multivariate positive-definite functions. Using these extensions and semidefinite programming can be improved some upper bounds for spherical codes.Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be defined for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approach to rational conformal field theories, we introduce geometric structure that is appropriate to define surface holonomy in more general situations: Jandl gerbes for unoriented surfaces, D-branes for surfaces with boundaries, and bi-branes for surfaces with defect lines.This article gives an overview of recent results on the relation between quantum field theory and motives, with an emphasis on two different approaches: a “bottom-up” approach based on the algebraic geometry of varieties associated to Feynman graphs, and a “top-down” approach based on the comparison of the properties of associated categorical structures. This survey is mostly based on joint work of the author with Paolo Aluffi, along the lines of the first approach, and on previous work of the author with Alain Connes on the second approach.