66 research outputs found
Hamiltonicity and -hypergraphs
We define and study a special type of hypergraph. A -hypergraph ), where is a partition of , is an
-uniform hypergraph having vertices partitioned into classes of
vertices each. If the classes are denoted by , ,...,, then a
subset of of size is an edge if the partition of formed by
the non-zero cardinalities , ,
is . The non-empty intersections are called the parts
of , and denotes the number of parts. We consider various types
of cycles in hypergraphs such as Berge cycles and sharp cycles in which only
consecutive edges have a nonempty intersection. We show that most
-hypergraphs contain a Hamiltonian Berge cycle and that, for and , a -hypergraph always contains a sharp
Hamiltonian cycle. We also extend this result to -intersecting cycles
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
The random k-matching-free process
Let be a graph property which is preserved by removal of edges,
and consider the random graph process that starts with the empty -vertex
graph and then adds edges one-by-one, each chosen uniformly at random subject
to the constraint that is not violated. These types of random
processes have been the subject of extensive research over the last 20 years,
having striking applications in extremal combinatorics, and leading to the
discovery of important probabilistic tools. In this paper we consider the
-matching-free process, where is the property of not
containing a matching of size . We are able to analyse the behaviour of this
process for a wide range of values of ; in particular we prove that if
or if then this process is likely to
terminate in a -matching-free graph with the maximum possible number of
edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds
on are essentially best possible, and we make a first step towards
understanding the behaviour of the process in the intermediate regime
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
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
Compositional Algorithms on Compositional Data: Deciding Sheaves on Presheaves
Algorithmicists are well-aware that fast dynamic programming algorithms are
very often the correct choice when computing on compositional (or even
recursive) graphs. Here we initiate the study of how to generalize this
folklore intuition to mathematical structures writ large. We achieve this
horizontal generality by adopting a categorial perspective which allows us to
show that: (1) structured decompositions (a recent, abstract generalization of
many graph decompositions) define Grothendieck topologies on categories of data
(adhesive categories) and that (2) any computational problem which can be
represented as a sheaf with respect to these topologies can be decided in
linear time on classes of inputs which admit decompositions of bounded width
and whose decomposition shapes have bounded feedback vertex number. This
immediately leads to algorithms on objects of any C-set category; these include
-- to name but a few examples -- structures such as: symmetric graphs, directed
graphs, directed multigraphs, hypergraphs, directed hypergraphs, databases,
simplicial complexes, circular port graphs and half-edge graphs.
Thus we initiate the bridging of tools from sheaf theory, structural graph
theory and parameterized complexity theory; we believe this to be a very
fruitful approach for a general, algebraic theory of dynamic programming
algorithms. Finally we pair our theoretical results with concrete
implementations of our main algorithmic contribution in the AlgebraicJulia
ecosystem.Comment: Revised and simplified notation and improved exposition. The
companion code can be found here:
https://github.com/AlgebraicJulia/StructuredDecompositions.j
Hamiltonicity below Dirac's condition
Dirac's theorem (1952) is a classical result of graph theory, stating that an
-vertex graph () is Hamiltonian if every vertex has degree at
least . Both the value and the requirement for every vertex to have
high degree are necessary for the theorem to hold.
In this work we give efficient algorithms for determining Hamiltonicity when
either of the two conditions are relaxed. More precisely, we show that the
Hamiltonian cycle problem can be solved in time , for some
fixed constant , if at least vertices have degree at least , or
if all vertices have degree at least . The running time is, in both
cases, asymptotically optimal, under the exponential-time hypothesis (ETH).
The results extend the range of tractability of the Hamiltonian cycle
problem, showing that it is fixed-parameter tractable when parameterized below
a natural bound. In addition, for the first parameterization we show that a
kernel with vertices can be found in polynomial time
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