Abstract
Let 
(G) denote the rectilinear crossing number of a graph G. We determine (K11)=102 and (K12)=153. Despite the remarkable hunt for crossing numbers of the complete graph K
n
– initiated by R. Guy in the 1960s – these quantities have been unknown forn>10 to date. Our solution mainly relies on a tailor-made method for enumerating all inequivalent sets of points (order types) of size 11.
Based on these findings, we establish a new upper bound on (K
n
) for general n. The bound stems from a novel construction of drawings of K
n
with few crossings.
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Aichholzer, O., Aurenhammer, F. & Krasser, H. On the Crossing Number of Complete Graphs. Computing 76, 165–176 (2006). https://doi.org/10.1007/s00607-005-0133-3
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DOI: https://doi.org/10.1007/s00607-005-0133-3