Article

K_6 minors in large 6-connected graphs

03/2012;
Source: arXiv

ABSTRACT Jorgensen conjectured that every 6-connected graph with no K_6 minor has a
vertex whose deletion makes the graph planar. We prove the conjecture for all
sufficiently large graphs.

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    ABSTRACT: It is shown that every sufficiently large almost-5-connected non-planar graph contains a minor isomorphic to an arbitrarily large graph from one of six families of graphs. The graphs in these families are also almost-5-connected, by which we mean that they are 4-connected and all 4-separations contain a “small” side. As a corollary, every sufficiently large almost-5-connected non-planar graph contains both a K3, 4-minor and a -minor. The connectivity condition cannot be reduced to 4-connectivity, as there are known infinite families of 4-connected non-planar graphs that do not contain a K3, 4-minor. Similarly, there are known infinite families of 4-connected non-planar graphs that do not contain a -minor. © 2012 Wiley Periodicals, Inc. (Contract grant sponsors: C8C Foundation; Inamori Foundation; Kayamori Foundation (K. K.).)
    Journal of Graph Theory 10/2012; 71(2):128-141. DOI:10.1002/jgt.20637 · 0.67 Impact Factor

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