The width of quadrangulations of the projective plane

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 displaystyle. Let G be a nonbipartite quadrangulation of the projective plane on n vertices. Our result immediately implies that G has edge-width at most , 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 , which is a constant away from the optimal; we prove a lower bound of . Finally, we show that G has an odd cycle transversal of size at most inducing a single edge, where is the maximum degree. This last result partially answers a question of Nakamoto and Ozeki.