Abstract
In a finite graph, an edge set Z is an element of the cycle space if and only if every vertex has even degree in Z. We extend this basic result to the topological cycle space, which allows infinite circuits, of locally finite graphs. In order to do so, it becomes necessary to attribute a parity to the ends of the graph.
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References
H. Bruhn: The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits, J. Combin. Theory (Series B) 92 (2004), 235–256.
H. Bruhn, R. Diestel and M. Stein: Cycle-cocycle partitions and faithful cycle covers for locally finite graphs, J. Graph Theory 50 (2005), 150–161.
H. Bruhn and M. Stein: MacLane’s planarity criterion for locally finite graphs, J. Combin. Theory (Series B) 96 (2006), 225–239.
H. Bruhn and M. Stein: On end degrees and infinite cycles in locally finite graphs, Combinatorica 27(3) (2007), 269–291.
Q. Cui, J. Wang and X. Yu: Hamilton circles in infinite planar graphs, J. Combin. Theory (Series B) 99 (2009), 110–138.
R. Diestel: Graph theory (3rd edition), Springer-Verlag, 2005.
R. Diestel and D. Kühn: On infinite cycles I, Combinatorica 24(1) (2004), 69–89.
R. Diestel and D. Kühn: On infinite cycles II, Combinatorica 24(1) (2004), 91–116.
R. Diestel and D. Kühn: Topological paths, cycles and spanning trees in infinite graphs; Europ. J. Combinatorics 25 (2004), 835–862.
A. Georgakopoulos: Infinite Hamilton cycles in squares of locally finite graphs, Adv. Math. 220(3) (2009), 670–705.
C. Thomassen and A. Vella: Graph-like continua, augmenting arcs, and Menger’s theorem; Combinatorica 28(5) (2008), 595–623.
A. Vella and R. B. Richter: Cycle spaces of topological spaces, J. Graph Theory 59 (2008), 115–144.