  Kuratowski proved that a finite graph embeds in the plane if it does not
contain a subdivision of either K_5 or K_{3,3}, called Kuratowski subgraphs. A
conjectured generalization of this result to all nonorientable surfaces says
that a finite minimal forbidden subgraph for the nonorientable surface of genus
g can be written as the union of g+1 Kuratowski subgraphs such that the union
of each pair of these fails to embed in the projective plane, the union of each
triple of these fails to embed in the Klein bottle if g >= 2, and the union of
each triple of these fails to embed in the torus if g >= 3. We show that this
conjecture is true for all minimal forbidden subgraphs of order < 10 for the
nonorientable surfaces of genus 3 and 4.

Combinatorics