Large non-planar graphs and an application to crossing-critical graphs

  • Authors:

  • Guoli Ding;Bogdan Oporowski;Robin Thomas;Dirk Vertigan

  • Affiliations:

  • Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA;Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA;Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2011

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Abstract

We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K"4","k, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3-connected 2-crossing-critical graphs with no subdivision isomorphic to the graph obtained from a cycle of length 2k by joining all pairs of diagonally opposite vertices.