# Veblen's theorem

In mathematics, **Veblen's theorem**, introduced by Oswald Veblen (1912), states that the set of edges of a finite graph can be written as a union of disjoint simple cycles if and only if every vertex has even degree. Thus, it is closely related to the theorem of Euler (1736) that a finite graph has an Euler tour (a single non-simple cycle that covers the edges of the graph) if and only if it is connected and every vertex has even degree. Indeed, a representation of a graph as a union of simple cycles may be obtained from an Euler tour by repeatedly splitting the tour into smaller cycles whenever there is a repeated vertex. However, Veblen's theorem applies also to disconnected graphs, and can be generalized to 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 adding more edges and vertices of *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).

## 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. - Sabidussi, Gert (1964), "Infinite Euler graphs",
*Canadian Journal of Mathematics*,**16**: 821–838, doi:10.4153/CJM-1964-078-x, MR 0169236. - Veblen, Oswald (1912), "An Application of Modular Equations in Analysis Situs",
*Annals of Mathematics*, Second Series,**14**(1): 86–94, doi:10.2307/1967604, JSTOR 1967604