Subdivisions of K5 in Graphs Embedded on Surfaces With Face-Width at Least 5

We prove that if G is a 5-connected graph embedded on a surface Sigma (other than the sphere) with face-width at least 5, then G contains a subdivision of K-5. This is a special case of a conjecture of P. Seymour, that every 5-connected nonplanar graph contains a subdivision of K-5. Moreover, we prove that if G is 6-connected and embedded with face-width at least 5, then for every v. V (G), G contains a subdivision of K5 whose branch vertices are v and four neighbors of v. (C) 2012 Wiley Periodicals, Inc.