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Series: Graph Theory Seminar

We discuss the relationship between the chromatic number (Chi),
the clique number (Omega) and maximum average degree (MAD).

Series: Graph Theory Seminar

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).

Series: Graph Theory Seminar

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.

Series: Graph Theory Seminar

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.

Series: Graph Theory Seminar

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)

Series: Graph Theory Seminar

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.

Series: Graph Theory Seminar

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.

Series: Graph Theory Seminar

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.

Series: Graph Theory Seminar

Grötzsch's theorem implies that every planar triangle-free graph is
3-colorable. It is natural to ask whether this can be improved. We prove that
every planar triangle-free graph on n vertices has fractional chromatic number
at most 3-1/(n+1/3), while Jones constructed planar triangle-free n-vertex
graphs with fractional chromatic number 3-3/(n+1). We also investigate additional
conditions under that triangle-free planar graphs have fractional chromatic
number smaller than 3-epsilon for some fixed epsilon > 0.(joint work with J.-S. Sereni and J. Volec)

Series: Graph Theory Seminar

Generalized triangle T_r is an r-graph with edges {1,2,…,r}, {1,2,…,r-1, r+1} and {r,r+1, r+2, …,2r-2}. The family \Sigma_r consists of all r-graphs with three edges D_1, D_2, D_3 such that |D_1\cap D_2|=r-1 and D_1\triangle D_2\subset D_3. In 1989 it was conjectured by Frankl and Furedi that ex(n,T_r) = ex(n,\Sigma_r) for large enough n, where ex(n,F) is the Tur\'{a}n function. The conjecture was proven to be true for r=3, 4 by Frankl, Furedi and Pikhurko respectively. We settle the conjecture for r=5,6 and show that extremal graphs are blow-ups of the unique (11, 5, 4) and (12, 6, 5) Steiner systems. The proof is based on a technique for deriving exact results for the Tur\'{a}n function from “local stability" results, which has other applications. This is joint work with Sergey Norin.