The set of irreducible graphs for the projective plane is finite

AbstractIn 1930 Kuratowski proved that a graph does not embed in the real plane R2 if and only if it contains a subgraph homeomorphic to one of two graphs, K5 or K3, 3. For positive integer n, let In (P) denote a smallest set of graphs whose maximal valency is n and such that any graph which does not embed in the real projective plane contains a subgraph homeomorphic to a graph in In (P) for some n. Glover and Huneke and Milgram proved that there are only 6 graphs in I3 (P), and Glover and Huneke proved that In (P) is finite for all n. This note proves that In (P) is empty for all but a finite number of n. Hence there is a finite set of graphs for the projective plane analogous to Kuratowski's two graphs for the plane