  We show that every $4$-chromatic graph on $n$ vertices, with no two
vertex-disjoint odd cycles, has an odd cycle of length at most
$\tfrac12\,(1+\sqrt{8n-7})$. Let $G$ be a non-bipartite quadrangulation of the
projective plane on $n$ vertices. Our result immediately implies that $G$ has
edge-width at most $\tfrac12\,(1+\sqrt{8n-7})$, which is sharp for infinitely
many values of $n$. We also show that $G$ has face-width (equivalently,
contains an odd cycle transversal of cardinality) at most $\tfrac14(1+\sqrt{16
n-15})$, which is a constant away from the optimal; we prove a lower bound of
$\sqrt{n}$. Finally, we show that $G$ has an odd cycle transversal of size at
most $\sqrt{2\Delta n}$ inducing a single edge, where $\Delta$ is the maximum
degree. This last result partially answers a question of Nakamoto and Ozeki.
