Large non-planar graphs and an application to crossing-critical graphs
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 K4,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 2. k by joining all pairs of diagonally opposite vertices. © 2010 Robin Thomas.
Publication Source (Journal or Book title)
Journal of Combinatorial Theory. Series B
Ding, G., Oporowski, B., Thomas, R., & Vertigan, D. (2011). Large non-planar graphs and an application to crossing-critical graphs. Journal of Combinatorial Theory. Series B, 101 (2), 111-121. https://doi.org/10.1016/j.jctb.2010.12.001