Definition of 2-factorable Graph Theory I would appreciate a simpler explanation of the following info
I am confused on what the definition of 2-factor is. It sounds like a perfect matching to me, but that's not what this paper is saying
 A: Ordinarily... a $k$-factor is a $k$-regular spanning subgraph.  Thus


*

*A $1$-factor is a perfect matching.

*A $2$-factor will be a spanning subgraph which is a union of disjoint cycles.
This is the ordinary definition used in, say, Petersen's $2$-factor Theorem.
So here's an example of a graph with a $2$-factor highlighted as colored cycles:

A $k$-factorization is a decomposition into $k$-factors.  A graph is $k$-factorable if it admits a $k$-factorization.
The above graph isn't $2$-factorable because it has vertices of odd degree.

Judging from the (now deleted) definition, the author:


*

*includes isolated edges as $2$-cycles, and

*defines a graph as "$2$-factorable" if there exists a $2$-factor for which after contracting the cycles, we obtain another graph with a $2$-factor.
According to this definition, a $1$-factor (perfect matching) would also be a $2$-factor.  And the above graph would be $2$-factorable.
I've never seen this definition before.
A: A graph $G$ is 2-factorable if and only if $G$ is $2k-regular$ for some positive integer $k$.
A perfect matching is 1-factorable. If $M$ is a perfect matching in a graph $G$, then $G[M]$ is a 1-regular spanning subgraph of $G$.
If you want a simple way of determining if a graph is 2-factorable, it must be able to be factored into Hamiltonian cycles.
