
> 
> Definition: A **$k$-factor** of a graph is a spanning $k$-regular
>  subgraph.
> 
> 
> Definition: A **$k$-factorization** of a graph is a partition of the edge
>  set into $k$-factors.
> 
> 
> 


Petersen's celebrated theorem says that a $2m$-regular graph has a $2$-factorization. A corollary is that a $2m$-regular graph also has a $2k$-factor for any $k \leq m$, since you can take the union of $k$ $2$-factors from the factorization.


My question is whether the converse of the corollary is true, that is - 



> 
> can every $2k$-factor of a $2m$-regular graph be written as a union of
>  $k$ factors from a $2$-factorization?
> 
> 
> 



Converse of Petersen's 2-Factorization Theorem
If I understand your question correctly, the answer is yes and follows directly from Petersen.


Let $H\_1$ be your $2k$-factor, and $H\_2$ the $(2m-2k)$-factor that remains of $G$ after deleting the edges of $H\_1$. Now apply Petersen to each of $H\_1$ and $H\_2$ to get $2$-factorizations in each, which combine to a $2$-factorization of $G$.


