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→2-factorization: The previous version stated that: "If a graph is 2-factorable, then it has to be 2k-regular for some integer k." This is directly contradicted by the Petersen graph, which is 3-regular and has a 2-factor. |
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==2-factorization==
If a connected graph is 2''k''-regular and has an even number of edges it may also be ''k''-factored, by choosing each of the two factors to be an alternating subset of the edges of an [[Euler tour]].<ref>{{harvtxt|Petersen|1891}}, §6, p. 198.</ref> This applies only to connected graphs; disconnected counterexamples include disjoint unions of odd cycles, or of copies of ''K''<sub>2''k''+1</sub>.
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