The crossing number of a projective graph is quadratic in the face–width

We show that for each integer g⩾0 there is a constant cg>0 such that every graph that embeds in the projective plane with sufficiently large face–width r has crossing number at least cgr2 in the orientable surface Σg of genus g. As a corollary, we give a polynomial time constant factor approximation algorithm for the crossing number of projective graphs with bounded degree.