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In mathematics, Veblen's theorem, introduced by , 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 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 . 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 . ==See also== *Cycle basis *Cycle double cover conjecture *Eulerian matroid 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Veblen's theorem」の詳細全文を読む スポンサード リンク
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