2-factorization
If a graph is 2-factorable, then it has to be 2k-regular for some integer k. Julius Petersen showed in 1891 that this necessary condition is also sufficient: any 2k-regular graph is 2-factorable.
If a connected graph is 2k-regular 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. This applies only to connected graphs; disconnected counterexamples include disjoint unions of odd cycles, or of copies of K2k+1.
Read more about this topic: Graph Factorization