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Keywords:

  • K_5;
  • subdivisions;
  • 5-connected;
  • face-width 5;
  • representativity 5

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 vV(G), G contains a subdivision of K5 whose branch vertices are v and four neighbors of v.