Large Non-Planar Graphs and an Application to Crossing-Critical Graphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Combinatorial Theory, Series B 101 (2011) 111–121 Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series B www.elsevier.com/locate/jctb Large non-planar graphs and an application to crossing-critical graphs Guoli Ding a,1,BogdanOporowskia, Robin Thomas b,2, Dirk Vertigan a,3 a Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA b School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA article info abstract Article history: We prove that, for every positive integer k, there is an integer Received 5 January 2010 N such that every 4-connected non-planar graph with at least N Available online 24 December 2010 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 Keywords: of vertices at distance exactly k, or the graph obtained from a Non-planar graph Crossing number cycle of length k by adding two vertices adjacent to each other Crossing-critical and to every vertex on the cycle. We also prove a version of this Minor for subdivisions rather than minors, and relax the connectivity to Subdivision 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.
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