I am under the impression that I have a simple misunderstanding of some terminology:
A 1-factor is defined to be a 1-regular spanning subgraph. So a 1-factor is a perfect matching. Yet a well-known result states that the Petersen graph is not 1-factorable. However, I can easily identify a perfect matching (many, in fact) of the Petersen graph.
I am assuming 1-factorable means 'possessing a 1-factor', or in other words, a graph is 1-factorable if and only if it has a perfect matching. Where has my terminology gone wrong so as to produce this apparent contradiction? Any assistance would be appreciated.
A $1$-factor is a $1$-regular spanning subgraph, but a $1$-factorization is a decomposition into edge-disjoint $1$-factors. A graph is $1$-factorable if it has a $1$-factorization.
The Petersen graph has some $1$-factors, but it does not have a $1$-factorization, because once you remove a $1$-factor (a perfect matchings), you will be left with some odd cycles (which do not, themselves, have perfect matchings). So the Petersen graph is not $1$-factorable.
(It does, of course, have a factorization into a $1$-factor and a $2$-factor: a perfect matching and a union of cycles.)