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We show that every 4‐chromatic graph on  n  vertices, with no two vertex‐disjoint odd cycles, has an odd cycle of length at most1  2(  1  +    8  n  −  7    )    . Let  G  be a nonbipartite quadrangulation of the projective plane on  n  vertices. Our result immediately implies that  G  has edge‐width at most1  2(  1  +    8  n  −  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 most1  4(  1  +    16  n  −  15    )    , which is a constant away from the optimal; we prove a lower bound of    n   . Finally, we show that  G  has an odd cycle transversal of size at most2  Δ  ninducing a single edge, where Δ is the maximum degree. This last result partially answers a question of Nakamoto and Ozeki.

The width of quadrangulations of the projective plane