Project
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Summary
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Details
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Contact
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Graphs with
ends: topological properties of the space |G|
DFG-gefördert |
The space |G| consisting of an infinite graph G and its ends is a normal space,
which is in general neither compact nor metrizable. Its connected
subsets need not be path-connected. The graphs G for which |G| is compact or metrizable can be
characterized; they include all locally finite graphs. The connected
subsets of |G| that are not
path-connected can also be characterized; such sets can exist when G is locally finite, but they
cannot be closed in |G|.
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Book
Survey
Papers
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Reinhard Diestel
Agelos
Georgakopoulos Philipp
Sprüssel |
Cycle space
theorems in locally finite graphs
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Most of the
classical theorems relating the cycle space of a finite graph to its
structural properties (such as planarity) fail for infinite graphs,
even for locally finite ones. However, if we build the cycle space of
those graphs G not just from
their (finite) cycles but from the edge sets of the topological circles
in their Freudenthal compactification |G|,
allowing
infinite
sums
when
they
are well-defined, we obtain a larger
space that performs much
better. This project aims to assess to what extent this new topological cycle space can assume
the structural role we have come to expect of the cycle space of a
graph.
So far, the evidence has been
overwhelmingly positive. Theorems found include infinite versions of
the Tutte/Kelmans planarity criterion, of Mac Lane's theorem, and of
plane duality and Whitney's theorem. As with a finite graph, the
topological cycle space is generated by its peripheral cycles, and by
its geodesic cycles, as long as the edge lengths chosen give rise to a
metric inducing the existing topology on |G|.
A major open conjecture is
that a set of edges lies in the cycle space of G if and only if induces even
degrees at all vertices and ends. (End degrees can be infinite in
locally finite graphs; their parity was defined by Bruhn and Stein.)
Other open problems currently investigated concern the topological bicycle space of G, the intersection of its
topological cycle space and its cut space.
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Survey
Papers
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Henning Bruhn
Reinhard
Diestel
Agelos
Georgakopoulos
Philipp
Sprüssel
Maya
Stein
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Menger's theorem for
infinite graphs with ends
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Menger's theorem, in the
form suggested by Erdös for infinite graphs, is proved for certain
graphs with ends, including all countable graphs. This means that the
sets of points to be linked may contain a mixture of vertices and ends.
The linkage is made up of paths – rays or double rays in the case of
linking ends – or, alternatively, arcs in the Freudenthal
compactification of the graph. |
Papers
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Henning Bruhn
Reinhard
Diestel
Maya
Stein |
Ramsey theory and
hypergraph regularity
GIF-gefördert
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Ramsey theory deals with
the existence of order within chaos. The basic paradigm is that when an
ordered object is partitioned into finitely many parts at least one of
these parts will contain an ordered substructure. The classical basic
example is that in any finite coloring of the edges of a sufficiently
large complete graph one will find a relatively large monochromatic
complete subgraph. In this project we study thresholds of Ramsey
properties in random hypergraphs and in random subsets of the integers.
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Paper |
Mathias Schacht
Yury
Person,
Hi&x1ec7;p Hàn |
