## Seminars and Colloquia by Series

Thursday, August 27, 2009 - 12:05 , Location: Skiles 255 , William T. Trotter , Math, GT , Organizer: Robin Thomas
Slightly modifying a quote of Paul Erdos: The problem for graphs we solve this week. The corresponding problem for posets will take longer. As one example, testing a graph to determine if it is planar is linear in the number of edges. Testing an order (Hasse) diagram to determine if it is planar is NP-complete. As a second example, it is NP-complete to determine whether a graph is a cover graph. With these remarks in mind, we present some results, mostly new but some classic, regarding posets with planar cover graphs and planar diagrams. The most recent result is that for every h, there is a constant c_h so that if P is a poset of height h and the cover graph of P is planar, then the dimension of P is at most c_h.
Thursday, June 11, 2009 - 11:05 , Location: Skiles 255 , Daniel Kral , ITI, Charles University, Prague , Organizer: Robin Thomas

We study several parameters of cubic graphs with large girth. In particular, we prove that every n-vertex cubic graph with sufficiently large girth satisfies the following:

• has a dominating set of size at most 0.29987n (which improves the previous bound of 0.32122n of Rautenbach and Reed)
• has fractional chromatic number at most 2.37547 (which improves the previous bound of 2.66881 of Hatami and Zhu)
• has independent set of size at least 0.42097n (which improves the previous bound of 0.41391n of Shearer), and
• has fractional total chromatic number arbitrarily close to 4 (which answers in the affirmative a conjecture of Reed). More strongly, there exists g such that the fractional total chromatic number of every bridgeless graph with girth at least g is equal to 4.
The presented bounds are based on a simple probabilistic argument.

The presentation is based on results obtained jointly with Tomas Kaiser, Andrew King, Petr Skoda and Jan Volec.

Thursday, June 4, 2009 - 11:00 , Location: Skiles 255 , Zdenek Dvorak , Simon Fraser University , Organizer: Robin Thomas
Richter and Salazar conjectured that graphs that are critical for a fixed crossing number k have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of k. We disprove these conjectures for every k >170, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree, and explore the structure of such graphs.
Thursday, April 23, 2009 - 12:05 , Location: Skiles 255 , Jie Ma , School of Mathematics, Georgia Tech , Organizer: Robin Thomas
A well know theorem of Kuratowski states that a graph is planar graph iff it contains no TK_5 or TK_{3,3}. In 1970s Seymour conjectured that every 5-connected nonplanar graph contains a TK_5. In the talk we will discuss several special cases of the conjecture, for example the graphs containing K_4^- (K_4 withour an edge). A related independent paths theorem also will be covered.
Wednesday, April 22, 2009 - 11:05 , Location: Skiles 269 , Matt Baker , School of Mathematics, Georgia Tech , Organizer: Robin Thomas
I will discuss some new results, as well as new interpretations of some old results, concerning reduced divisors (a.k.a. G-parking functions) on graphs, metric graphs, and tropical curves.
Thursday, March 19, 2009 - 11:00 , Location: Skiles 269 , Sergey Norin , Princeton University , Organizer: Robin Thomas
Many results in asymptotic extremal combinatorics are obtained using just a handful of instruments, such as induction and Cauchy-Schwarz inequality. The ingenuity lies in combining these tools in just the right way. Recently, Razborov developed a flag calculus which captures many of the available techniques in pure form, and allows one, in particular, to computerize the search for the right combination. In this talk we outline the general approach and describe its application to the conjecture that a digraph with minimum outdegree n/3 contains a directed triangle. This special case of the Caccetta-Haggkvist conjecture has been extensively investigated in the past. We show that a digraph with minimum outdegree a*n contains a directed triangle for a = 0.3465. The proof builds on arguments used to establish previously known bounds, due to Shen from 1998 (a = 0.3542) and Hamburger, Haxell and Kostochka from 2008 (a = 0.3531). It consists of a combination of ~80 computer generated inequalities. Based on joint work with Jan Hladky and Daniel Kral.
Thursday, February 19, 2009 - 12:05 , Location: Skiles 255 , Peter Horak , University of Washington, Tacoma , Organizer: Robin Thomas
Tiling problems belong to the oldest problems in whole mathematics. They attracted attention of many famous mathematicians. Even one of the Hilbert problems is devoted to the topic. The interest in tilings by unit cubes originated with a conjecture raised by Minkowski in 1908. In this lecture we will discuss the conjecture, and other closely related problems.
Thursday, January 15, 2009 - 12:00 , Location: Skiles 255 , Hein van der Holst , Eindhoven University of Technology , Organizer: Robin Thomas
Each graph can be embedded in 3-space. The problem becomes more interesting if we put restrictions on the type of embedding. For example, a linkless embedding of a graph is one where each pair of vertex-disjoint circuits has linking number equal to zero. The class of all graphs that have a linkless embedding is closed under taking minors. Robertson, Seymour, and Thomas gave the forbidden minors for this class of graphs. Open remained how to find a linkless embedding in polynomial time. In the talk we start with discussing an algorithm to find a linkless embedding.Instead of embedding the graph in 3-space, we could also consider mapping properties of certain superstructures of the graph in 3-space, and, indeed, if this superstructure has not the right mapping properties in 3-space, see whether it has the right one in 4-space, etc. Recently, we introduced for a graph G a new graph parameter \sigma(G), which is defined as the smallest d such that superstructures of G have a zero intersection mapping in d-space. The nicest property of this graph parameter is its independence of the superstructure and thus depends on the graph only. For d=2 and d=3, \sigma(G) \leq d if and only if G is outerplanar and planar, respectively. The graphs G with \sigma(G)\leq 4 are exactly those that have a linkless embedding. In the second part of the talk we will discuss this new graph parameter. (This part is joint work with R. Pendavingh.)
Thursday, November 20, 2008 - 12:05 , Location: Skiles 255 , Carl Yerger , School of Mathematics, Georgia Tech , Organizer: Robin Thomas
Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that given any configuration of k pebbles on G and any specified vertex v in V(G), there is a sequence of pebbling moves that sends a pebble to v. We will show that the pebbling number of a graph of diameter four on n vertices is at most 3n/2 + O(1), and this bound is best possible up to an additive constant. This proof, based on a discharging argument and a decomposition of the graph into ''irreducible branches'', generalizes work of Bukh on graphs of diameter three. Further, we prove that the pebbling number of a graph on n vertices with diameter d is at most (2^{d/2} - 1)n + O(1). This also improves a bound of Bukh.
Thursday, October 30, 2008 - 11:05 , Location: Skiles 255 , Stavros Garoufalidis , School of Mathematics, Georgia Tech , Organizer: Robin Thomas

PLEASE NOTE UNUSUAL TIME

We will consider the problem of counting the number T(n,g) of cubic graphs with n edges on a surface of of genus g, and review was is known in the combinatorial community in the past 30 years, what was conjectured in physics 20 years ago, and what was proven last month in joint work with Thang Le and Marcos Marino, using the Riemann-Hilbert analysis of the Painleve equation. No knowledge of physics or analysis is required.