## The Cycle Space of an Infinite Graph (2004)

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Venue: | COMB., PROBAB. COMPUT |

Citations: | 33 - 7 self |

### BibTeX

@ARTICLE{Diestel04thecycle,

author = {Reinhard Diestel},

title = {The Cycle Space of an Infinite Graph},

journal = {COMB., PROBAB. COMPUT},

year = {2004},

volume = {14},

pages = {2005}

}

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### Abstract

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach

### Citations

133 |
On the problem of decomposing a graph into n connected factors
- Tutte
- 1961
(Show Context)
Citation Context ... with finite k, 7 even locally finite ones, although this had been conjectured by Nash-Williams [25]. (See [1] for another counterexample.) What Tutte thought about this is not recorded. In his paper =-=[33]-=-, he does not claim to have an infinite counterexample to the finite statement, but he does prove alocally finite version of a weaker statement. When read by itself, this weaker statement appears awkw... |

100 |
A theorem on planar graphs
- Tutte
- 1956
(Show Context)
Citation Context ... of our approach to higher-dimensional complexes. 19sPerhaps the most prominent problem of the first kind above is to extend Tutte’s theorem that 4-connected finite planar graphs have Hamilton cycle=-=s [32]-=-. Nash-Williams [26] conjectured that an infinite 4-connected planar graph contains a spanning double ray unless it has more than two ends. 8 This restriction is clearly necessary. But maybe it is jus... |

69 | Graph Theory (2nd edition - Diestel - 2000 |

64 |
2-Linked graphs
- Thomassen
- 1980
(Show Context)
Citation Context ...es in at most two peripheral circuits. Recall that G ∗ =(V ∗ ,E ∗ )isadual of G =(V,E) ifE ∗ = E and, for every F ⊆ E, the set F is a circuit in G if and only if F is a minimal cut in G ∗ =-=. Thomassen [29, 30] stu-=-died the weaker notion of a dual where this is asked only of finite sets F ⊆ E; let us call such a graph G ∗ a finitary dual of G. In our context, though, where infinite sets of edges can be circu... |

58 | Edge-disjoint spanning trees of finite graphs - Nash-Williams - 1961 |

48 |
Decomposition of finite graphs into forests
- Nash-Williams
- 1964
(Show Context)
Citation Context ...for arbitrary countable graphs. A problem of the opposite kind, where the naive non-topological extension of a finite result is true but too easy, is the following. Nash-Williams’s arboricity theore=-=m [24] says th-=-at a finite graph has an edge-partition into at most k forests if and only if every set of ℓ ≥ 1vertices induces at most k (ℓ − 1) edges. (The condition is obviously necessary.) Since an infin... |

40 |
Über die Enden topologischer Räume und Gruppen
- Freudenthal
(Show Context)
Citation Context .... In fact, there are several that fit the bill, of which one is clearly better than the others [10]. To define this topology (which is not at all new: its origins go back to Jung [17] and Freudenthal =-=[15]), consider-=- a graph G =(V,E) with its set Ω=Ω(G) ofends. Let G itself carry the topology of a 1-complex. 2 To extend this topology to Ω, let us define for each end ω ∈ Ωabasis of open neighbourhoods. ... |

36 | On infinite cycles
- Diestel, Kühn
(Show Context)
Citation Context ...ibilities for an ‘extremal’ branch of infinite graph theory. Numerous open problems are suggested. Introduction This is an expository paper describing a new line of research started jointly with K=-=ühn [10, 11, 12] a-=-nd with Bruhn and Stein [4], some independent work of the latter two authors inspired by this approach [2, 5, 6], and some further observations not yet published. Our aim here will be to • describe ... |

28 | Topological paths, cycles and spanning trees in infinite graphs
- Diestel, Kühn
(Show Context)
Citation Context ...ibilities for an ‘extremal’ branch of infinite graph theory. Numerous open problems are suggested. Introduction This is an expository paper describing a new line of research started jointly with K=-=ühn [10, 11, 12] a-=-nd with Bruhn and Stein [4], some independent work of the latter two authors inspired by this approach [2, 5, 6], and some further observations not yet published. Our aim here will be to • describe ... |

25 | Graph-theoretical versus topological ends of graphs - Diestel, Kuhn - 2003 |

24 | A combinatorial condition for planar graphs - MacLane - 1937 |

24 |
Planarity and duality of finite and infinite graphs
- Thomassen
- 1980
(Show Context)
Citation Context ...es in at most two peripheral circuits. Recall that G ∗ =(V ∗ ,E ∗ )isadual of G =(V,E) ifE ∗ = E and, for every F ⊆ E, the set F is a circuit in G if and only if F is a minimal cut in G ∗ =-=. Thomassen [29, 30] stu-=-died the weaker notion of a dual where this is asked only of finite sets F ⊆ E; let us call such a graph G ∗ a finitary dual of G. In our context, though, where infinite sets of edges can be circu... |

19 | MacLane’s planarity criterion for locally finite graphs
- Bruhn, Stein
(Show Context)
Citation Context ...is an expository paper describing a new line of research started jointly with Kühn [10, 11, 12] and with Bruhn and Stein [4], some independent work of the latter two authors inspired by this approach=-= [2, 5, 6], -=-and some further observations not yet published. Our aim here will be to • describe the main ideas underlying this new approach, introducing the new concepts by motivating examples before defining t... |

17 | On end degrees and infinite circuits in locally finite graphs, Combinatorica, to appear - Bruhn, Stein |

16 |
The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits
- Bruhn
(Show Context)
Citation Context ...is an expository paper describing a new line of research started jointly with Kühn [10, 11, 12] and with Bruhn and Stein [4], some independent work of the latter two authors inspired by this approach=-= [2, 5, 6], -=-and some further observations not yet published. Our aim here will be to • describe the main ideas underlying this new approach, introducing the new concepts by motivating examples before defining t... |

16 |
Cycle-cocycle partitions and faithful cycle covers for locally graphs
- Bruhn, Diestel, et al.
(Show Context)
Citation Context ...infinite graph theory. Numerous open problems are suggested. Introduction This is an expository paper describing a new line of research started jointly with Kühn [10, 11, 12] and with Bruhn and Stein=-= [4], -=-some independent work of the latter two authors inspired by this approach [2, 5, 6], and some further observations not yet published. Our aim here will be to • describe the main ideas underlying thi... |

13 |
The end structure of a graph: recent results and open problems
- Diestel
- 1992
(Show Context)
Citation Context ...ith infinite degrees the statement is easily seen to be false. Another intrinsic problem concerns our new notion of a topological spanning tree and their relationship to ‘end-faithful’ spanning tr=-=ees [7]. Th-=-e existence problem for end-faithful spanning trees has a long history and was 22seventually settled negatively [18, 28, 31]. However, all the known ‘minimal’ counterexamples are infinitely connec... |

10 |
Graph theory, 2nd edition, Springer-Verlag 2000 and http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/download.html
- Diestel
(Show Context)
Citation Context ...well be viewed as a very surprising topological characterization of the graphs that admit a dual: 5sTheorem 1.9 (Whitney 1933) G has a dual if and only if it is planar. By colouring-flow duality (see =-=[8]-=-), the four colour theorem can be rephrased as follows: Theorem 1.10 (4CT) If G is planar and bridgeless, then E is a union of two elements of C. The last theorem in our list is also intimately linked... |

8 | Normal spanning trees, Aronszajn trees and excluded minors
- Diestel, Leader
(Show Context)
Citation Context ...panning tree? We remark that the closure in |G| of a normal spanning tree of G is always a topological spanning tree in |G|. The connected graphs that have normal spanning trees were characterized in =-=[14]-=-; they include all countable connected graphs. Finally, readers with a topological background may have wondered about possible generalizations of our approach to higher dimensions. Indeed, it is not d... |

8 |
Connectivity in infinite graphs
- Jung
- 1971
(Show Context)
Citation Context ...y can indeed be found. In fact, there are several that fit the bill, of which one is clearly better than the others [10]. To define this topology (which is not at all new: its origins go back to Jung =-=[17] and Freude-=-nthal [15]), consider a graph G =(V,E) with its set Ω=Ω(G) ofends. Let G itself carry the topology of a 1-complex. 2 To extend this topology to Ω, let us define for each end ω ∈ Ωabasis of ... |

7 |
On a packing problem for infinite graphs and independence
- Oxley
- 1979
(Show Context)
Citation Context ...heorem 5.3, we can extend Theorem 1.10: Theorem 5.10 [3] If G is bridgeless and planar, then E is a union of two elements of C. It remains to generalize the Tutte/Nash-Williams packing theorem. Oxley =-=[27]-=- showed that Theorem 1.11 does not extend to infinite graphs with finite k, 7 even locally finite ones, although this had been conjectured by Nash-Williams [25]. (See [1] for another counterexample.) ... |

5 |
Miscellaneous problems on infinite graphs
- Halin
(Show Context)
Citation Context ...’ before ‘set’ throughout), complete with proofs. Curiously, though, none of Theorems 1.4–1.11 remains true. As a case in point, let us look at Theorem 1.7. Here are two counterexamples, due t=-=o Halin [16]-=- and Bruhn [2] respectively. In the graph of Figure 1, D Figure 1: E(C) isnot a finite sum of peripheral circuits the edge set of the separating cycle C is not a finite sum (mod 2) of peripheral circu... |

5 |
An end-faithful spanning tree counterexample
- Seymour, Thomas
- 1991
(Show Context)
Citation Context ...opological spanning tree and their relationship to ‘end-faithful’ spanning trees [7]. The existence problem for end-faithful spanning trees has a long history and was 22seventually settled negativ=-=ely [18, 28, 31]. Ho-=-wever, all the known ‘minimal’ counterexamples are infinitely connected, and in particular have only one end. For such graphs, however, it is easy to find a topological spanning tree: start with a... |

4 |
Infinite, highly connected digraphs with no two arc-disjoint spanning trees
- Aharoni, Thomassen
- 1989
(Show Context)
Citation Context ...tudy the circles in terms of their circuits, as planned. (For future reference we remark that the corresponding statements hold also for arcs in |G|, the homeomorphic images of the real unit interval =-=[0,1]-=-.) 2 Every edge is homeomorphic to the real interval [0, 1], the basic open sets around an inner point being just the open intervals on the edge. The basic open neighbourhoods of avertex x are the uni... |

4 |
Infinite connected graphs with no end-preserving spanning trees
- Thomassen
- 1992
(Show Context)
Citation Context ...opological spanning tree and their relationship to ‘end-faithful’ spanning trees [7]. The existence problem for end-faithful spanning trees has a long history and was 22seventually settled negativ=-=ely [18, 28, 31]. Ho-=-wever, all the known ‘minimal’ counterexamples are infinitely connected, and in particular have only one end. For such graphs, however, it is easy to find a topological spanning tree: start with a... |

3 | Decompositions of graphs into closed and endless chains - Nash-Williams - 1960 |

2 |
Unexplored and semi-explored territories in graph theory
- Nash-Williams
- 1973
(Show Context)
Citation Context ...higher-dimensional complexes. 19sPerhaps the most prominent problem of the first kind above is to extend Tutte’s theorem that 4-connected finite planar graphs have Hamilton cycles [32]. Nash-William=-=s [26]-=- conjectured that an infinite 4-connected planar graph contains a spanning double ray unless it has more than two ends. 8 This restriction is clearly necessary. But maybe it is just an indication that... |

2 | Cycle-cocycle partitions, treepacking and arboricity in locally finite graphs, preprint 2004. http://www.math.uni-hamburg.de/math/research/preprints/hbm.html - Bruhn, Diestel, et al. |

2 | End spaces and spanning trees, preprint - Diestel - 2004 |

1 |
Characterizing the cycle space of a locally finite graph by vertex and end degrees, in preparation
- Bruhn, Stein
(Show Context)
Citation Context ...is an expository paper describing a new line of research started jointly with Kühn [10, 11, 12] and with Bruhn and Stein [4], some independent work of the latter two authors inspired by this approach=-= [2, 5, 6], -=-and some further observations not yet published. Our aim here will be to • describe the main ideas underlying this new approach, introducing the new concepts by motivating examples before defining t... |

1 |
Martin’s axiom and spanning trees of infinite graphs
- Komjáth
- 1992
(Show Context)
Citation Context ...opological spanning tree and their relationship to ‘end-faithful’ spanning trees [7]. The existence problem for end-faithful spanning trees has a long history and was 22seventually settled negativ=-=ely [18, 28, 31]. Ho-=-wever, all the known ‘minimal’ counterexamples are infinitely connected, and in particular have only one end. For such graphs, however, it is easy to find a topological spanning tree: start with a... |

1 |
Decompositions of infinite graphs
- Laviolette
- 2003
(Show Context)
Citation Context ...4 to these graphs: a topological Euler tour exists if and only if all vertices and all ends have even degree. G1 Figure 8: G1 has a topological Euler tour; G2 does not Building on ideas of Laviolette =-=[19]-=-, Bruhn and Stein [6] succeeded in generalizing this idea into a comprehensive definition for the degrees of ends that makes the simultaneous extensions of Theorem 1.4 and (1.1) possible. Given an end... |

1 |
Infinite graphs –asurvey
- Nash-Williams
- 1967
(Show Context)
Citation Context ...te/Nash-Williams packing theorem. Oxley [27] showed that Theorem 1.11 does not extend to infinite graphs with finite k, 7 even locally finite ones, although this had been conjectured by Nash-Williams =-=[25]-=-. (See [1] for another counterexample.) What Tutte thought about this is not recorded. In his paper [33], he does not claim to have an infinite counterexample to the finite statement, but he does prov... |

1 |
Infinite Paths in Planar Graphs I–III, preprints
- Yu
- 1999
(Show Context)
Citation Context ...finitely many components. This implies that any infinite part left by a finite separator contains a ray, so the two versions are equivalent. A proof of Nash-Williams’s conjecture has been given by Y=-=u [35]. -=-20sA more fundamental consequence of taking the ends of a graph into account when describing its structure, and in particular of the new notion of end degrees, might be the feasibility of an ‘extrem... |

1 | Duality in infinite graphs, preprint 2004 - Bruhn, Diestel |

1 | On infinite cycles II, Combinatorica, (to appear). http://www.math.uni-hamburg.de/math/research/preprints/hbm.html 10 See Freudenthal [14], or [12], for the definition of ends in arbitrary topological spaces, and conditions on those spaces that ensure add - Diestel, Kühn |

1 | On infinite cycles I, Combinatorica 24 - Diestel, Kühn - 2004 |

1 | Infinite Paths in Planar Graphs I–V, preprints 1999-2004. Version 27.10.2004 - Yu |