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The Kelmans-Seymour conjecture

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The Kelmans-Seymour conjecture
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24
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Seymour and, independently, Kelmans conjectured that every 5-connected non-planar graph contains a subdivision of $K_5$. We have recently proved this conjecture. I will give a sketch of our proof, and mention several related problems. Joint work with D. He and Y. Wang.
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