Subdivisions of K5 in Graphs Embedded on Surfaces With Face-Width at Least 5
Article first published online: 24 OCT 2012
© 2012 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 74, Issue 2, pages 182–197, October 2013
How to Cite
Krakovski, R., Stephens, D. C. and Zha, X. (2013), Subdivisions of K5 in Graphs Embedded on Surfaces With Face-Width at Least 5. J. Graph Theory, 74: 182–197. doi: 10.1002/jgt.21700
- Issue published online: 2 AUG 2013
- Article first published online: 24 OCT 2012
- Manuscript Revised: 24 JAN 2012
- Manuscript Received: 28 SEP 2010
- face-width 5;
- representativity 5
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.