Quadrangulations and 5-critical Graphs on the Projective Plane

Abstract

Let Q be a nonbipartite quadrangulation of the projective plane. Youngs [You96] proved that Q cannot be 3-colored. We prove that for every 4-coloring of Q and for any two colors a and b, the number of faces F of Q, on which all four colors appear and colors a and b are not adjacent on F, is odd. This strengthens previous results that have appeared in [You96, HRS02, Moh02, CT04]. If we form a triangulation of the projective plane by inserting a vertex of degree 4 in every face of Q, we obtain an Eulerian triangulation T of the projective plane whose chromatic number is 5. The above result shows that T is never 5-critical. We show that sometimes one can remove two, three, or four, vertices from T and obtain a 5-critical graph. This gives rise to an explicit construction of 5-critical graphs on the projective plane and yields the first explicit family of 5-critical graphs with arbitrarily large edge-width on a fixed surface.