Definition: A set $C⊆V$ and a collection of subsets $_1,…,_⊆V$ is an odd vertex cover if for every edge either $∩≠∅$ or $ ⊆ B_i$ for some .
The cost of the odd vertex cover is defined to be: $|C| + \sum_{i=1}^{k} \lfloor{\frac{|B_i|}{2}}\rfloor$
Without loss of generality, we may assume that all the sets $B_i$ are of odd size, hence the name, and that C and $B_1,…,B_k$ are disjoint.
Theorem: In any graph, the size of the maximum matching is equal to the cost of the minimum odd vertex cover.
How can I prove it? I tried to use Edmond's algorithm for finding maximum match. Then to take one side of these edges as $C$ and the blossoms as $B_i$'s. I don't have a formal proof for that though.