Basics of Pfaffians

Let $G(V,E)$ be an undirected graph. An orientation of $G$ is Pfaffian if every even cycle $C$ such that $G \setminus V(C)$ has a perfect matching, has an odd number of edges directed in either direction of the cycle. $G$ is called Pfaffian if it has at least one such orientation.

• Prove that $K_{3,3}$ is not Pfaffian.

• Prove that the Petersen graph is not Pfaffian.