Subdivisions of K5 in Graphs Embedded on Surfaces With Face‐Width at Least 5. Issue 2 (24th October 2012) - Krakovski, Roi; Stephens, D. Christopher; Zha, Xiaoya
Abstract We prove that if G is a 5‐connected graph embedded on a surface Σ (other than the sphere) with face‐width at least 5, then G contains a subdivision of K5. This is a special case of a conjecture of P. Seymour, that every 5‐connected nonplanar graph contains a subdivision of K5. 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.