Exercise 2, § 2

Let $G = (S,F, o, t)$ be a quiver. One says that G is a bipartite quiver if there exists a partition $P =\{S_1,S_2\}$ into subsets of S such that for every arrow $f\in F$, the origin $o(f)$ and the term $t(f)$ belong to different parts of the partition P.

a) Suppose that G is connected. Prove that there exists at most one partition P as above.

b) Prove that a graph G is bipartite if and only if every loop in G is of even length.