Infinite graphs with ends: a topological approach
All Hamburg and some other papers on this topic
The original topological approach group in 2007
The best starting point is perhaps the introductory but
- Locally finite graphs with ends: a topological approach IIII
(R. Diestel), Discrete Math 311312 (201011); PDF
of parts III
While the survey is more comprehensive (and includes many
pointers to what might be interesting to look at next, including
countless open problems), it is also written in a less formal
style that makes slightly more demands on the reader. The book
chapter may help with precise definitions, should the survey be
found to be too informal. It also offers a selection of proofs of
basic facts, which are typical for this area and make good
introductory reading. There is also an older expository text,
mostly written around 2002:
- The cycle space of an infinite graph (R. Diestel), Comb.
Probab. Computing 14 (2005), 59-79; PDF
The first few sections of this contain a lot of motivation for
the topological concepts used in this field, and still have some
General properties of the topological space formed by a graph
and its ends:
- Tangles and the Stone-Čech compactification of infinite graphs (J. Kurkofka & M. Pitz), preprint 2018; ArXiv.
- Graph-like compacta: characterizations and eulerian loops (B.
Espinoza, P. Gartside & M. Pitz), preprint 2016; ArXiv.
- Ends and tangles (R. Diestel), Abhandlungen Math. Sem. Univ.
Hamburg 2017; PDF
- Brownian motion on graph-like spaces (A. Georgakopoulos &
K. Kolesko), preprint 2013; PDF
- On graph-like continua of finite length (A. Georgakopoulos),
Topol. Appl. (2014), 188-208; PDF
- Dual trees must share their ends (R. Diestel & J.
J. Combin. Theory (Series B) 123 (2017) 32-53; PDF
- Graph topologies induced by edge lengths (A. Georgakopoulos),
Discrete Math. 311 (special issue 2011), 1523-1542; PDF.
- The fundamental group of a locally finite graph with ends (R.
Diestel & P. Sprüssel), Advances Math. 226 (2011),
- End spaces of graphs are normal (P. Sprüssel), J. Combin.
Theory (Series B), 98 (2008), 798-804; PDF.
- Duality of ends (H. Bruhn & M. Stein), Combinatorics,
Probability and Computing, 12 (2009), 47-60; PDF.
- End spaces and spanning trees (R. Diestel), J. Combin. Theory
(Series B) 96 (2006), 846-854; abstract;
- Connected but not path-connected subspaces of infinite graphs
(A. Georgakopoulos), Combinatorica 27 No.6 (2007), 683-698; PDF.
- Graph-theoretical versus topological ends of graphs (R.
Diestel & D. Kühn), J. Combin. Theory (Series B) 87 (2003),
Homology / Cycle space:
- Orthogonality and minimality in the homology of locally finite
graphs (R. Diestel & J. Pott), Electronic J. Comb. 21
(2014), #P3.36; PDF.
- Cycle decompositions: from graphs to continua (A.
Georgakopoulos), Advances Math. 229 (2012), 935-967; ArXiv
- On the homology of locally compact spaces with ends (R.
Diestel & P. Sprüssel), Topology and its Applications
(2011), to appear; PDF
- Eulerian edge sets in locally finite graphs (E. Berger &
H. Bruhn), Combinatorica 31 (2011), 21-38; PDF
- Topological circles and Euler tours in locally finite graphs
(A. Georgakopoulos), Electronic J. Comb. 16:#R40 (2009); PDF
- Bicycles and left-right tours in locally finite graphs (H.
Bruhn, S. Kosuch & M. Win Myint), Europ. J. Combinatorics 30
(2009), 356-371; PDF.
- The homology of locally finite graphs with ends (R. Diestel
& P. Sprüssel), Combinatorica 30 (2010), 681-714; abstract;
- Bases and closed spaces with infinite sums (H. Bruhn & A.
Georgakopoulos), Linear Algebra and its Applications (2011), to
- Geodetic topological cycles in locally finite graphs (A.
Georgakopoulos & P. Sprüssel), Electronic J. Comb.
16:#R144 (2009); PDF
- Duality in infinite graphs (H. Bruhn & R. Diestel), Comb.
Probab. Computing 15 (2006), 75-90; abstract;
- On end degrees and infinite cycles in locally finite graphs
(H. Bruhn & M. Stein), Combinatorica 27 (2007),
- MacLane's planarity criterion for locally finite graphs (H.
Bruhn & M. Stein), J. Combin. Theory (Series B) 96
(2006), 225-239; PDF.
- Der Zyklenraum nicht lokal-endlicher Graphen (M. Schulz),
Diplomarbeit Hamburg 2005, PDF.
- Cycle-cocycle partitions and faithful cycle covers for locally
finite graphs (H. Bruhn, R. Diestel & M. Stein),
J. Graph Theory 50 (2005), 150-161; abstract;
- Topological paths, cycles and spanning trees in infinite
graphs (R. Diestel & D. Kühn), Europ. J. Combinatorics 25
(2004), 835-862; abstract;
- The cycle space of a 3-connected locally finite graph is
generated by its finite and infinite peripheral circuits (H.
Bruhn), JCTB 92 (2004), 235-256; PDF.
- On infinite cycles I (R. Diestel & D. Kühn), Combinatorica
24 (2004), 68-89; abstract;
- On infinite cycles II (R. Diestel & D. Kühn),
Combinatorica 24 (2004), 91-116; abstract;
Extremal infinite graph theory:
- An analogue of Edmonds' Branching Theorem for infinite
digraphs (J.P. Gollin & K. Heuer), preprint 2018; ArXiv
- Hamilton cycles in infinite cubic graphs (M. Pitz), to appear
in Electron. J. Comb. ArXiv
- Hamiltonicity in locally finite graphs: two extensions and a
counterexample (K. Heuer); PDF
- Contractible edges in 2-connected locally finite graphs (Tsz
Lung Chan), Electronic J. Comb 22 (2015) #P2.47; PDF
- A sufficient local degree condition for the hamiltonicity of
locally finite claw-free graphs (K. Heuer), Europ.J.Comb. 55
(2016), 82-99; PDF
- Extending cycles locally to Hamilton cycles (M. Hamann, F.
Lehner, J. Pott), preprint 2013; PDF
- A sufficient condition for Hamiltonicity in locally finite
graphs (K. Heuer), Europ. J. Combinatorics 45 (2015), 97-114; PDF
- Forcing finite minors in sparse infinite graphs by
large-degree assumptions (R. Diestel), to appear in Electronic
J. Combinatorics; PDF
- Extremal infinite graph theory (survey) (M. Stein), to appear
in the Infinite Graph Theory
special volume of Discrete Math (2011); PDF
- Ends and vertices of small degree in infinite minimally
k-(edge-)connected graphs (M. Stein), preprint 2009; PDF
- Infinite Hamilton cycles in squares of locally finite graphs
(A. Georgakopoulos), Advances Math., 220 (2009), 670-705; PDF
- Forcing highly connected subgraphs in locally finite graphs
(M. Stein), J. Graph Theory 54 (2007), 331-349; PDF
- Arboriticity and tree-packing in locally finite graphs (M.
Stein), J. Combin. Theory (Series B) 96 (2006), 302-312; PDF.
- Hamilton cycles in planar locally finite graphs (H. Bruhn
& X. Yu), SIAM. J. Discrete Math. 22 (2008), 1381-1392; PDF
Some theses in this area:
in directed and undirected infinite graphs (K. Heuer,
simply connected 2-complexes in 3-space..., (J. Carmesin,
in finite and infinite graphs (J. Carmesin, 2015)
length induces end topologies (T. Rühmann, 2014)
line graph of every locally finite 6-edge-connected graph with
finitely many ends is hamiltonian, (F. Lehner,
graphs with a tree-like structure, (M. Hamann, 2011)
Flüsse (T. Rühmann, Bachelorarbeit 2010)
- On the homology of
infinite graphs with ends (P. Sprüssel, 2010)
questions in graph theory (M. Stein, Habilitationsschrift
and their circuits: from finite to infinite (H. Bruhn,
and left-right tours in locally finite graphs (M. Win
paths and cycles in infinite graphs (A. Georgakopoulos,
Zyklenraum nicht lokal-endlicher Graphen (M. Schulz,
of graphs (M. Stein, 2005)
circuits in locally finite graphs (H. Bruhn, 2005)
highly connected planar graphs of large girth
(A.Georgakopoulos, Diplomarbeit 2004)
- Cycles, minors and trees
(D. Kühn, 2001)