Hi,

this is how i understand your language, pls correct me if otherwise:

"odd vertex" means vertex of odd degree,

"size of G" is the number of vertices of G.

Then the first statement is obvious (partition of the graph consists of the set of odd-degree vertices and the set of even-degree vertices),

and the second statement is wrong ( consider graph G=(V,E) where V={u,v,w} and E={{u,v},{v,w}} ).