Thursday, September 19, 2013 - 12:05 , Location: Skiles 005 , Paul Wollan , School of Mathematics, Georgia Tech and University of Rome, Italy , Organizer: Robin Thomas
While Robertson and Seymour showed that graphs are well-quasi-ordered under the minor relation, it is well known that directed graphs are not. We will present an exact characterization of the minor-closed sets of directed graphs which are well-quasi-ordered. This is joint work with M. Chudnovsky, S. Oum, I. Muzi, and P. Seymour.
Tuesday, September 10, 2013 - 12:05 , Location: Skiles 005 , Martin Loebl , Charles University , Organizer: Robin Thomas
We prove the dcdc conjecture in a class of lean fork graphs, argue that this class is substantial and show a path towards the complete solution. Joint work with Andrea Jimenez.
Thursday, April 18, 2013 - 12:05 , Location: Skiles 005 , Daniel Kral , University of Warwick , Organizer: Robin Thomas
A systematic study of large combinatorial objects has recently led to discovering many connections between discrete mathematics and analysis. In this talk, we apply analytic methods to permutations. In particular, we associate every sequence of permutations with a measure on a unit square and show the following: if the density of every 4-element subpermutation in a permutation p is 1/4!+o(1), then the density of every k-element subpermutation is 1/k!+o(1). This answers a question of Graham whether quasirandomness of a permutation is captured by densities of its 4-element subpermutations. The result is based on a joint work with Oleg Pikhurko.
Thursday, April 4, 2013 - 12:05 , Location: Skiles 005 , Dhruv Mubayi , University of Illinois at Chicago , Organizer: Robin Thomas
Since the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs over 20 years ago, there has been a lot of effort by many researchers to extend the theory to hypergraphs. I will present some of this history, and then describe our recent results that provide such a generalization and unify much of the previous work. One key new aspect in the theory is a systematic study of hypergraph eigenvalues first introduced by Friedman and Wigderson. This is joint work with John Lenz.
Thursday, March 28, 2013 - 12:05 , Location: Skiles 005 , Peter Whalen , Georgia Tech , Organizer: Robin Thomas
We show that any internally 4-connected non-planar bipartite graph contains a subdivision of K3,3 in which each subdivided path contains an even number of vertices. In addition to being natural, this result has broader applications in matching theory: for example, finding such a subdivision of K3,3 is the first step in an algorithm for determining whether or not a bipartite graph is Pfaffian. This is joint work with Robin Thomas.
Tuesday, March 26, 2013 - 12:05 , Location: Skiles 005 , Bernard Lidicky , University of Illinois at Urbana-Champaign , Organizer: Robin Thomas
A recent lower bound on the number of edges in a k-critical n-vertex graph by Kostochka and Yancey yields a half-page proof of the celebrated Grotzsch Theorem that every planar triangle-free graph is 3-colorable. We use the same bound to give short proofs of other known theorems on 3-coloring of planar graphs, among whose is the Grunbaum-Aksenov Theorem that every planar with at most three triangles is 3-colorable. We also prove the new result that every graph obtained from a triangle-free planar graph by adding a vertex of degree at most four is 3-colorable. Joint work with O. Borodin, A. Kostochka and M. Yancey.
Thursday, March 14, 2013 - 12:05 , Location: Skiles 005 , Vojtech Tuma , Charles University , Organizer: Robin Thomas
We present a dynamic data structure representing a graph G, which allows addition and removal of edges from G and can determine the number of appearances of a graph of a bounded size as an induced subgraph of G. The queries are answered in constant time. When the data structure is used to represent graphs from a class with bounded expansion (which includes planar graphs and more generally all proper classes closed on topological minors, as well as many other natural classes of graphs with bounded average degree), the amortized time complexity of updates is polylogarithmic. This data structure is motivated by improving time complexity of graph coloring algorithms and of random graph generation.
Thursday, February 21, 2013 - 12:05 , Location: Skiles 005 , Zdenek Dvorak , Charles University and Georgia Tech , Organizer: Robin Thomas
Every subcubic triangle-free graph on n vertices contains an independent set of size at least 5n/14 (Staton'79). We strengthen this result by showing that all such graphs have fractional chromatic number at most 14/5, thus confirming a conjecture by Heckman and Thomas. (Joint work with J.-S. Sereni and J. Volec)
Tuesday, February 19, 2013 - 12:05 , Location: Skiles 005 , Debmalya Panigrahi , Duke University , Organizer: Prasad Tetali
The online matching problem has received significant attention in recent years because of its connections to allocation problems in internet advertising, crowd sourcing, etc. In these real-world applications, the typical goal is not to maximize the number of allocations; rather it is to maximize the number of “successful” allocations, where success of an allocation is governed by a stochastic event that comes after the allocation. These applications motivate us to introduce stochastic rewards in the online matching problem. In this talk, I will formally define this problem, point out its connections to previously studied allocation problems, give a deterministic algorithm that is close to optimal in its competitive ratio, and describe some directions of future research in this line of work. (Based on joint work with Aranyak Mehta.)
Thursday, February 14, 2013 - 12:05 , Location: Skiles 005 , Paul Wollan , University of Rome and Georgia Tech , Organizer: Robin Thomas
The Weak Structure Theorem of Robertson and Seymour is the cornerstone of many of the algorithmic applications of graph minors techniques. The theorem states that any graph which has both large tree-width and excludes a fixed size clique minor contains a large, nearly planar subgraph. In this talk, we will discuss a new proof of this result which is significantly simpler than the original proof of Robertson and Seymour. As a testament to the simplicity of the proof, one can extract explicit constants to the bounds given in the theorem. We will assume no previous knowledge about graph minors or tree-width. This is joint work with Ken Kawarabayashi and Robin Thomas