78 research outputs found
Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given
Fast winning strategies in Avoider-Enforcer games
In numerous positional games the identity of the winner is easily determined.
In this case one of the more interesting questions is not {\em who} wins but
rather {\em how fast} can one win. These type of problems were studied earlier
for Maker-Breaker games; here we initiate their study for unbiased
Avoider-Enforcer games played on the edge set of the complete graph on
vertices. For several games that are known to be an Enforcer's win, we
estimate quite precisely the minimum number of moves Enforcer has to play in
order to win. We consider the non-planarity game, the connectivity game and the
non-bipartite game
Covering and tiling hypergraphs with tight cycles
Given , we say that a -uniform hypergraph is a
tight cycle on vertices if there is a cyclic ordering of the vertices of
such that every consecutive vertices under this ordering form an
edge. We prove that if and , then every -uniform
hypergraph on vertices with minimum codegree at least has
the property that every vertex is covered by a copy of . Our result is
asymptotically best possible for infinitely many pairs of and , e.g.
when and are coprime.
A perfect -tiling is a spanning collection of vertex-disjoint copies
of . When is divisible by , the problem of determining the
minimum codegree that guarantees a perfect -tiling was solved by a
result of Mycroft. We prove that if and is not divisible
by and divides , then every -uniform hypergraph on vertices
with minimum codegree at least has a perfect
-tiling. Again our result is asymptotically best possible for infinitely
many pairs of and , e.g. when and are coprime with even.Comment: Revised version, accepted for publication in Combin. Probab. Compu
Combinatorics, Probability and Computing
The main theme of this workshop was the use of probabilistic
methods in combinatorics and theoretical computer science. Although
these methods have been around for decades, they are being refined all
the time: they are getting more and more sophisticated and powerful.
Another theme was the study of random combinatorial structures,
either for their own sake, or to tackle extremal questions. The workshop
also emphasized connections between probabilistic combinatorics and
discrete probability
Topics in Extremal and Probabilistic Combinatorics.
PhD ThesisThis thesis encompasses several problems in extremal and probabilistic combinatorics.
Chapter 1. Tuza's famous conjecture on the saturation number states that for r-uniform
hypergraphs F the value sat(F; n)=nr1 converges. I answer a question of Pikhurko
concerning the asymptotics of the saturation number for families of hypergraphs, proving
in particular that sat(F; n)=nr1 need not converge if F is a family of r-uniform
hypergraphs.
Chapter 2. Cern y's conjecture on the length of the shortest reset word of a synchronizing
automaton is arguably the most long-standing open problem in the theory of nite
automata. We consider the minimal length of a word that resets some k-tuple. We
prove that for general automata if this is nite then it is
nk1
. For synchronizing
automata we improve the upper bound on the minimal length of a word that resets some
triple.
Chapter 3. The existence of perfect 1-factorizations has been studied for various families
of graphs, with perhaps the most famous open problem in the area being Kotzig's conjecture
which states that even-order complete graphs have a perfect 1-factorization. In my
work I focus on another well-studied family of graphs: the hypercubes. I answer almost
fully the question of how close (in some particular sense) to perfect a 1-factorization of
the hypercube can be.
Chapter 4. The k-nearest neighbour random geometric graph model puts vertices randomly
in a d-dimensional box and joins each vertex to its k nearest neighbours. I nd
signi cantly improved upper and lower bounds on the threshold for connectivity for the
k-nearest neighbour graph in high dimensions.
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