8 research outputs found
End spaces of graphs are normal
We show that the topological space of any infinite graph and its ends is
normal. In particular, end spaces themselves are normal.Comment: 8 page
Approximating infinite graphs by normal trees
We show that every connected graph can be approximated by a normal tree, up
to some arbitrarily small error phrased in terms of neighbourhoods around its
ends. The existence of such approximate normal trees has consequences of both
combinatorial and topological nature.
On the combinatorial side, we show that a graph has a normal spanning tree as
soon as it has normal spanning trees locally at each end; i.e., the only
obstruction for a graph to having a normal spanning tree is an end for which
none of its neighbourhoods has a normal spanning tree.
On the topological side, we show that the end space , as well as
the spaces naturally associated with a graph , are
always paracompact. This gives unified and short proofs for a number of results
by Diestel, Spr\"ussel and Polat, and answers an open question about
metrizability of end spaces by Polat.Comment: 9 page
Countably determined ends and graphs
The directions of an infinite graph are a tangle-like description of its
ends: they are choice functions that choose compatibly for all finite vertex
sets a component of . Although every direction is
induced by a ray, there exist directions of graphs that are not uniquely
determined by any countable subset of their choices. We characterise these
directions and their countably determined counterparts in terms of star-like
substructures or rays of the graph. Curiously, there exist graphs whose
directions are all countably determined but which cannot be distinguished all
at once by countably many choices. We structurally characterise the graphs
whose directions can be distinguished all at once by countably many choices,
and we structurally characterise the graphs which admit no such countably many
choices. Our characterisations are phrased in terms of normal trees and
tree-decompositions. Our four (sub)structural characterisations imply
combinatorial characterisations of the four classes of infinite graphs that are
defined by the first and second axiom of countability applied to their end
spaces: the two classes of graphs whose end spaces are first countable or
second countable, respectively, and the complements of these two classes.Comment: 17 pages, 2 figure
Weak geodesic topology and fixed finite subgraph theorems in infinite partial cubes I. Topologies and the geodesic convexity
AbstractThe weak geodesic topology on the vertex set of a partial cube G is the finest weak topology on V(G) endowed with the geodesic convexity. We prove the equivalence of the following properties: (i) the space V(G) is compact; (ii) V(G) is weakly countably compact; (iii) the vertex set of any ray of G has a limit point; (iv) any concentrated subset of V(G) (i.e. a set A such that any two infinite subsets of A cannot be separated by deleting finitely many vertices) has a finite positive number of limit points. Moreover, if V(G) is compact, then it is scattered. We characterize the partial cubes for which the weak geodesic topology and the geodesic topology (see [N. Polat, Graphs without isometric rays and invariant subgraph properties I. J. Graph Theory27 (1998), 99–109]) coincide, and we show that the class of these particular partial cubes is closed under Cartesian products, retracts and gated amalgams
A representation theorem for end spaces of infinite graphs
End spaces of infinite graphs sit at the interface between graph theory,
group theory and topology. They arise as the boundary of an infinite graph in a
standard sense generalising the theory of the Freudenthal boundary developed by
Freudenthal and Hopf in the 1940's for infinite groups.
A long-standing quest in infinite graph theory with a rich body of literature
seeks to describe the possible end structures of graphs by a set of
low-complexity representatives. In this paper we present a solution to this
fifty-year-old problem by showing that every end space is homeomorphic to the
end space of some (uniform graph on a) special order tree.Comment: 23 pages. V2 adds a moreover-part to Theorem 3.
A tree-of-tangles theorem for infinite tangles
Carmesin has extended Robertson and Seymour's tree-of-tangles theorem to the
infinite tangles of locally finite infinite graphs. We extend it further to the
infinite tangles of all infinite graphs.
Our result has a number of applications for the topology of infinite graphs,
such as their end spaces and their compactifications.Comment: 30 pages, 7 figure