A refinement of Kuratowski's theorem

Dedicated to the memory of Gabriel Andrew Dirac
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Abstract

The conjecture of Kelmans that any 3-connected non-planar graph with at least six vertices contains a cycle with three pairwise crossing chords is proved. Using this, a refinement of Kuratowski's theorem which also includes the result of Tutte that a graph is planar if and only if every cycle has a bipartite overlap graph is obtained.

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