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Series: Combinatorics Seminar

I will divide the talk between two topics. The first is Stirling numbers of the second kind, $S(n,k)$. For each $n$ the maximum $S(n,k)$ is achieved either at a unique $k=K_n$, or is achieved twice consecutively at $k=K_n,K_n+1$. Call those $n$ of the second kind {\it exceptional}. Is $n=2$ the only exceptional integer? The second topic is $m\times n$ nonnegative integer matrices all of whose rows sum to $s$ and all of whose columns sum to $t$, $ms=nt$. We have an asymptotic formula for the number of these matrices, valid for various ranges of $(m,s;n,t)$. Although obtained by a lengthy calculation, the final formula is succinct and has an interesting probabilistic interpretation. The work presented here is collaborative with Carl Pomerance and Brendan McKay, respectively.

Series: Combinatorics Seminar

Let a_1,...,a_k satisfy a_1+...+a_k=1 and suppose a k-uniform hypergraph on n
vertices satisfies the following property; in any partition of its vertices into k
sets A_1,...,A_k of sizes a_1*n,...,a_k*n, the number of edges intersecting
A_1,...,A_k is the number one would expect to find in a random k-uniform hypergraph.
Can we then infer that H is quasi-random? We show that the answer is negative if and
only if a_1=...=a_k=1/k. This resolves an open problem raised in 1991 by Chung and
Graham [J. AMS '91].
While hypergraphs satisfying the property corresponding to a_1=...=a_k=1/k are not
necessarily quasi-random, we manage to find a characterization of the hypergraphs
satisfying this property. Somewhat surprisingly, it turns out that (essentially)
there is a unique non quasi-random hypergraph satisfying this property. The proofs
combine probabilistic and algebraic arguments with results from the theory of
association schemes.
Joint work with Raphy Yuster

Series: Combinatorics Seminar

The map coloring problem is one of the major catalysts of the tremendous
development of graph theory. It was observed by Tutte that the problem of
the face-coloring of an planar graph can be formulated in terms of integer
flows of the graph. Since then the topic of integer flow has been one of the
most attractive in graph theory. Tutte had three famous fascinating flow
conjectures: the 3-flow conjecture, the 4-flow conjecture and the 5-flow
conjecture. There are some partial results for these three conjectures. But
in general, all these 3 conjectures are open.
Group connectivity is a generalization of integer flow of graphs. It
provides us with contractible flow configurations which play an important
role in reducing the graph size for integer flow problems, it is also
related to all generalized Tutte orientations of graphs. In this talk, I
will present an introduction and survey on group connectivity of graphs as
well as some open problems in this field.

Series: Combinatorics Seminar

One of the biggest hurdles in high performance computing today is the analysis of
massive quantities of data. As the size of the datasets grows to petascale (and beyond),
new techniques are needed to efficiently compute meaningful information from the raw data.
Graph-based data (which is ubiquitous in social networks, biological interaction networks, etc) poses additional challenges due to the difficulty of parallelizing many common graph algorithms. A key component in success is the generation of "realistic" random data sets for testing and benchmarking new algorithms.
The R-MAT graph generator introduced by Chakrabarti, Faloutsos, and Zhan (2004) offers a simple, fast method for generating very large directed graphs. One commonly held belief regarding graphs produced by R-MAT
is that they are "scale free"; in other words, their degree distribution follows a
power law as is observed in many real world networks. These properties have made R-MAT a popular choice for generating graphs for use in a variety of research disciplines including graph theoretic benchmarks,
social network analysis, computational biology, and network monitoring.
However, despite its wide usage and elegant, parsimonius design, our recent work
provides the first rigorous mathematical analysis of the degree distributions of
the generated graphs. Applying results from occupancy problems in probability theory, we
derive exact expressions for the degree distributions and other parameters.
We also prove that in the limit (as the number of vertices tends to
infinity), graphs generated with R-MAT have degree distributions that can be expressed as a
mixture of normal distributions. This talk will focus on the techniques used in solving this applied problem in terms of classical "ball and urn" results, including a minor extension of Chistyakov's theorem.

Series: Combinatorics Seminar

Hajos' conjecture is false, and it seems that graphs without a
subdivision of a big complete graph do not behave as well as
those without a minor of a big complete graph.
In fact, the graph minor theorem (a proof of Wagner's
conjecture) is not true if we replace the minor relation by the
subdivision relation. I.e, For every infinite sequence
G_1,G_2, ... of graphs, there exist distinct integers
i < j such that G_i is a minor of G_j, but if we replace
''minor" by ''subdivision", this is no longer true.
This is partially because we do not really know what the graphs
without a subdivision of a big complete graph look like.
In this talk, we shall discuss this issue. In particular,
assuming some moderate connectivity condition, we can say
something, which we will present in this talk.
Topics also include coloring graphs without a subdivision of a
large complete graph, and some algorithmic aspects. Some of the
results are joint work with Theo Muller.

Series: Combinatorics Seminar

Linkage involves finding a set of internally disjoint paths in a graph with specified endpoints. Given graphs G and H, we say G is H-linked if for every injective mapping f:V(H) -> V(G) we can find a subgraph H' of G which is a subdivision of H, with f(v) being the vertex of H' corresponding to each vertex v of H. We describe two results on H-linkage for small graphs H.

(1) Goddard showed that 4-connected planar triangulations are 4-ordered, or in other words C_4-linked. We strengthen this by showing that 4-connected planar triangulations are (K_4-e)-linked.

(2) Xingxing Yu characterized certain graphs related to P_4-linkage. We use his characterization to show that every 7-connected graph is P_4-linked, and to construct 6-connected graphs that are not P_4-linked.

This is joint work with Michael D. Plummer and Gexin Yu.

Series: Combinatorics Seminar

Stability methods are often used in extremal graph theory, Ramsey theory and similar areas, where an extremal problem is to be solved and

- we have a conjecture about the structure of the conjectured extremal configurations and according to our conjecture, it has some given property \mathcal P;
- we can prove that all the almost extremal structures are near to the property \mathcal P, in some sense;
- if we knew that if a structure is near to the property \mathcal P and is extremal, then it is already the conjectured structure.

Of course, stability methods can also be used in other cases, but we restrict ourselves to the above two areas.

In my lecture I will give an introduction to the applications of the stability methods in extremal graph theory, describe cases in extremal graph theory, extremal hypergraph theory, in the Erdos-Frankl-Rold (= generalized Erdos-Kleitman-Rothschild theory) ...

In the second part of my lecture I shall describe the application of this method to the Erdos-Sos conjecture. This is part of our work with Ajtai, Komlos and Szemeredi.

Series: Combinatorics Seminar

This is joint work with Dr. Yi Zhao.

Graph tiling problems can be summarized as follows: given a graph H, what conditions do we need to find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. The most familiar example of a graph tiling problem is finding a matching in a graph. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove a degree condition that guarantees an arbitrary bipartite graph G will be tiled by an arbitrary bipartite graph H. We also prove this degree condition is best possible up to a constant. This answers a question of Zhao and proves an asymptotic version of a result of Kuhn and Osthus for bipartite graphs.

Series: Combinatorics Seminar

There has been substantial work on approximation algorithms for clustering
data under distance-based objective functions such as k-median, k-means, and
min-sum objectives. This work is fueled in part by the hope that
approximating these objectives well will indeed yield more accurate
solutions. That is, for problems such as clustering proteins by function, or
clustering images by subject, there is some unknown correct "target"
clustering and the implicit assumption is that clusterings that are
approximately optimal in terms of these distance-based measures are also
approximately correct in terms of error with respect to the target. In this
work we show that if we make this implicit assumption explicit -- that is, if
we assume that any c-approximation to the given clustering objective Phi is
epsilon-close to the target -- then we can produce clusterings that are
O(epsilon)-close to the target, even for values c for which obtaining a
c-approximation is NP-hard. In particular, for the k-median, k-means, and
min-sum objectives, we show that we can achieve this guarantee for any
constant c > 1.
Our results show how by explicitly considering the alignment between the
objective function used and the true underlying clustering goals, one can
bypass computational barriers and perform as if these objectives were
computationally substantially easier.
This talk is based on joint work with Avrim Blum and Anupam Gupta (SODA
2009), Mark Braverman (COLT 2009), and Heiko Roeglin and Shang-Hua Teng (ALT 2009).

Series: Combinatorics Seminar

In this talk I will discuss a new technique discovered by myself
and Olof Sisask which produces many new insights in additive combinatorics,
not to mention new
proofs of classical theorems previously proved only using harmonic
analysis. Among these new proofs is one for Roth's theorem on three-term
arithmetic progressions, which gives the best bounds so
far achieved by any combinatorial method. And another is a new proof
that positive density subsets of the integers mod p contain very
long arithmetic progressions, first proved by Bourgain, and improved
upon by Ben Green and Tom Sanders. If time permits, I will discuss
how the method can be applied to the 2D corners problem.