Bipartite graphs

$(\Rightarrow)$. In a bipartite graph with parts $X,Y$, any cycle alternates $X$-$Y$-$X$-$Y$, so its length is even.

$(\Leftarrow)$. Suppose no odd cycles. Color each component as follows: pick a root, color it red; color each neighbor blue; propagate. If a conflict occurs (two adjacent vertices of the same color), we find a path between them of opposite parity, giving an odd cycle.

Conclusion. Bipartite $\iff$ no odd cycle.