
Crossing numbers
The Crossing Number of the Complete Graph ★★★
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The crossing number of
is the minimum number of crossings in all drawings of
in the plane.

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The Crossing Number of the Complete Bipartite Graph ★★★
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The crossing number of
is the minimum number of crossings in all drawings of
in the plane.

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The Crossing Number of the Hypercube ★★
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The crossing number of
is the minimum number of crossings in all drawings of
in the plane.
The -dimensional (hyper)cube
is the graph whose vertices are all binary sequences of length
, and two of the sequences are adjacent in
if they differ in precisely one coordinate.

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Drawing disconnected graphs on surfaces ★★
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Crossing sequences ★★
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Then there exists a graph that be drawn on a surface with orientable (nonorientable, resp.) genus with
crossings, but not with less crossings.
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Crossing numbers and coloring ★★★
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We let denote the crossing number of a graph
.



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Are different notions of the crossing number the same? ★★★
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![\[ \text{pair-cr}(G) = \text{cr}(G) \]](/files/tex/8cece1e00bb0e9fc122e0a5cad0dab2681cf33a4.png)
The crossing number of a graph
is the minimum number of edge crossings in any drawing of
in the plane. In the pairwise crossing number
, we minimize the number of pairs of edges that cross.
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