Veblen's theorem
In mathematics, Veblen's theorem, introduced by
infinite graphs in which every vertex has finite degree (Sabidussi 1964
).
If a countably infinite graph G has no odd-degree vertices, then it may be written as a union of disjoint (finite) simple cycles if and only if every finite subgraph of G can be extended (by including more edges and vertices from G) to a finite Eulerian graph. In particular, every countably infinite graph with only one end and with no odd vertices can be written as a union of disjoint cycles (Sabidussi 1964).
See also
- Cycle basis
- Cycle double cover conjecture
- Eulerian matroid
References
- Euler, L. (1736), "Solutio problematis ad geometriam situs pertinentis" (PDF), Commentarii Academiae Scientiarum Imperialis Petropolitanae, 8: 128–140. Reprinted and translated in Biggs, N. L.; Lloyd, E. K.; Wilson, R. J. (1976), Graph Theory 1736–1936, Oxford University Press.
- MR 0169236.
- JSTOR 1967604