## Seminars and Colloquia by Series

Thursday, October 1, 2015 - 13:30 , Location: Skiles 006 , Jie Ma , University of Science and Technology of China , Organizer: Xingxing Yu
There has been extensive research on cycle lengths in graphs with large minimum degree. In this talk, we will present several new and tight results in this area. Let G be a graph with minimum degree at least k+1. We prove that if G is bipartite, then there are k cycles in G whose lengths form an arithmetic progression with common difference two. For general graph G, we show that G contains \lfloor k/2\rfloor cycles with consecutive even lengths, and in addition, if G is 2-connected and non-bipartite, then G contains \lfloor k/2\rfloor cycles with consecutive odd lengths. Thomassen (1983) made two conjectures on cycle lengths modulo a fixed integer k: (1) every graph with minimum degree at least k+1 contains cycles of all even lengths modulo k; (2) every 2-connected non-bipartite graph with minimum degree at least $k+1$ contains cycles of all lengths modulo k. These two conjectures, if true, are best possible. Our results confirm both conjectures! when k is even. And when k is odd, we show that minimum degree at least \$+4 suffices. Moreover, our results derive new upper bounds of the chromatic number in terms of the longest sequence of cycles with consecutive (even or odd) lengths. This is a joint work with Chun-Hung Liu.
Thursday, September 10, 2015 - 13:35 , Location: Skiles 005 , Chun-Hung Liu , Princeton University , Organizer: Robin Thomas
A set F of graphs has the Erdos-Posa property if there exists a function f such that every graph either contains k disjoint subgraphs each isomorphic to a member in F or contains at most f(k) vertices intersecting all such subgraphs. In this talk I will address the Erdos-Posa property with respect to three closely related graph containment relations: minor, topological minor, and immersion. We denote the set of graphs containing H as a minor, topological minor and immersion by M(H),T(H) and I(H), respectively. Robertson and Seymour in 1980's proved that M(H) has the Erdos-Posa property if and only if H is planar. And they left the question for characterizing H in which T(H) has the Erdos-Posa property in the same paper. This characterization is expected to be complicated as T(H) has no Erdos-Posa property even for some tree H. In this talk, I will present joint work with Postle and Wollan for providing such a characterization. For immersions, it is more reasonable to consider an edge-variant of the Erdos-Posa property: packing edge-disjoint subgraphs and covering them by edges. I(H) has no this edge-variant of the Erdos-Posa property even for some tree H. However, I will prove that I(H) has the edge-variant of the Erdos-Posa property for every graph H if the host graphs are restricted to be 4-edge-connected. The 4-edge-connectivity cannot be replaced by the 3-edge-connectivity.
Thursday, April 16, 2015 - 12:05 , Location: Skiles 005 , Luke Postle , University of Waterloo , Organizer: Robin Thomas
We discuss the relationship between the chromatic number (Chi), the clique number (Omega) and maximum average degree (MAD).
Thursday, April 9, 2015 - 12:05 , Location: Skiles 005 , Torsten Muetze , School of Mathematics, Georgia Tech and ETH Zurich , Organizer: William T. Trotter
For integers k>=1 and n>=2k+1, the bipartite Kneser graph H(n,k) is defined as the graph that has as vertices all k-element and all (n-k)-element subsets of {1,2,...,n}, with an edge between any two vertices (=sets) where one is a subset of the other. It has long been conjectured that all bipartite Kneser graphs have a Hamilton cycle. The special case of this conjecture concerning the Hamiltonicity of the graph H(2k+1,k) became known as the 'middle levels conjecture' or 'revolving door conjecture', and has attracted particular attention over the last 30 years. One of the motivations for tackling these problems is an even more general conjecture due to Lovasz, which asserts that in fact every connected vertex-transitive graph (as e.g. H(n,k)) has a Hamilton cycle (apart from five exceptional graphs). Last week I presented a (rather technical) proof of the middle levels conjecture. In this talk I present a simple and short proof that all bipartite Kneser graphs H(n,k) have a Hamilton cycle (assuming that H(2k+1,k) has one). No prior knowledge will be assumed for this talk (having attended the first talk is not a prerequisite). This is joint work with Pascal Su (ETH Zurich).
Thursday, March 5, 2015 - 00:05 , Location: Skiles 005 , Ruidong Wang , Math, GT , Organizer: Robin Thomas
In the combinatorics of posets, many theorems are in pairs, one for chains and one for antichains. Typically, the statements are exactly the same when roles are reversed, but the proofs are quite different. The classic pair of theorems due to Dilworth and Mirsky were the starting point for this pattern, followed by the more general pair known respectively as the Greene-Kleitman and Greene theorems dealing with saturated partitions. More recently, a new pair has been discovered dealing with matchings in the comparability and incomparability graphs of a poset. We show that if the dimension of a poset P is d and d is at least 3, then there is a matching of size d in the comparability graph of P, and a matching of size d in the incomparability graph of P.
Tuesday, January 20, 2015 - 12:05 , Location: Skiles 006 , Martin Loebl , Charles University , Organizer: Robin Thomas
We express weight enumerator of each binary linear code as a product. An analogous result was obtain by R. Feynman in the beginning of 60's for the speacial case of the cycle space of the planar graphs.
Thursday, January 8, 2015 - 12:05 , Location: Skiles 005 , Andrea Jimenez , GT and University of São Paulo , Organizer: Robin Thomas
We discuss a dual version of a problem about perfect matchings in cubic graphs posed by Lovász and Plummer. The dual version is formulated as follows: "Every triangulation of an orientable surface has exponentially many groundstates"; we consider groundstates of the antiferromagnetic Ising Model. According to physicist, the dual formulation holds. In this talk, I plan to show a counterexample to the dual formulation (**), a method to count groundstates which gives a better bound (for the original problem) on the class of Klee-graphs, the complexity of the related problems and if time allows, some open problems. (**): After that physicists came up with an explanation to such an unexpected behaviour!! We are able to construct triangulations where their explanation fails again. I plan to show you this too. (This is joint work with Marcos Kiwi)
Wednesday, December 3, 2014 - 15:05 , Location: Skiles 005 , Zdenek Dvorak , Charles University , Organizer: Robin Thomas
For relations {R_1,..., R_k} on a finite set D, the {R_1,...,R_k}-CSP is a computational problem specified as follows: Input: a set of constraints C_1, ..., C_m on variables x_1, ..., x_n, where each constraint C_t is of form R_{i_t}(x_{j_{t,1}}, x_{j_{t,2}}, ...) for some i_t in {1, ..., k} Output: decide whether it is possible to assign values from D to all the variables so that all the constraints are satisfied. The CSP problem is boolean when |D|=2. Schaefer gave a sufficient condition on the relations in a boolean CSP problem guaranteeing its polynomial-time solvability, and proved that all other boolean CSP problems are NP-complete. In the planar variant of the problem, we additionally restrict the inputs only to those whose incidence graph (with vertices C_1, ..., C_m, x_1, ..., x_m and edges joining the constraints with their variables) is planar. It is known that the complexities of the planar and general variants of CSP do not always coincide. For example, let NAE={(0,0,1),(0,1,0),(1,0,0),(1,1,0),(1,0,1),(0,1,1)}). Then {NAE}-CSP is NP-complete, while planar {NAE}-CSP is polynomial-time solvable. We give some partial progress towards showing a characterization of the complexity of planar boolean CSP similar to Schaefer's dichotomy theorem.Joint work with Martin Kupec.
Thursday, August 28, 2014 - 13:30 , Location: Skiles 005 , Bartosz Walczak , GT, Math and Jagiellonian University in Krakow , Organizer: Robin Thomas
The dimension of a poset P is the minimum number of linear extensions of P whose intersection is equal to P. This parameter plays a similar role for posets as the chromatic number does for graphs. A lot of research has been carried out in order to understand when and why the dimension is bounded. There are constructions of posets with height 2 (but very dense cover graphs) or with planar cover graphs (but unbounded height) that have unbounded dimension. Streib and Trotter proved in 2012 that posets with bounded height and with planar cover graphs have bounded dimension. Recently, Joret et al. proved that the dimension is bounded for posets with bounded height whose cover graphs have bounded tree-width. My current work generalizes both these results, showing that the dimension is bounded for posets of bounded height whose cover graphs exclude a fixed (topological) minor. The proof is based on the Robertson-Seymour and Grohe-Marx structural decomposition theorems. I will survey results relating the dimension of a poset to structural properties of its cover graph and present some ideas behind the proof of the result on excluded minors.
Thursday, August 21, 2014 - 13:30 , Location: Skiles 005 , Chun-Hung Liu , Math, GT and Princeton University , Organizer: Robin Thomas
Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation. In other words, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). Unlike the relation of minor and weak immersion, the topological minor relation does not well-quasi-order graphs in general. However, Robertson conjectured in the late 1980s that for every positive integer k, the topological minor relation well-quasi-orders graphs that do not contain a topological minor isomorphic to the path of length k with each edge duplicated. We will sketch the idea of our recent proof of this conjecture. In addition, we will give a structure theorem for excluding a fixed graph as a topological minor. Such structure theorems were previously obtained by Grohe and Marx and by Dvorak, but we push one of the bounds in their theorems to the optimal value. This improvement is needed for our proof of Robertson's conjecture. This work is joint with Robin Thomas.