The set of irreducible graphs for the projective plane is finite

Abstract

In 1930 Kuratowski proved that a graph does not embed in the real plane R 2 if and only if it contains a subgraph homeomorphic to one of two graphs, K 5 or K 3, 3. For positive integer n, let I n ( 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 I n ( P) for some n. Glover and Huneke and Milgram proved that there are only 6 graphs in I 3 ( P), and Glover and Huneke proved that I n ( P) is finite for all n. This note proves that I n ( 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.