 
Summary: On the Crossing Number of Complete Graphs
O. Aichholzer , Graz, and F. Aurenhammer, Graz,
and H. Krasser y , Graz
June 3, 2005
Abstract
Let cr(G) denote the rectilinear crossing number of a graph G. We
determine cr(K 11 ) = 102 and cr(K 12 ) = 153. Despite the remark
able hunt for crossing numbers of the complete graph K n { initiated
by R. Guy in the 1960s { these quantities have been unknown for
n > 10 to date. Our solution mainly relies on a tailormade method
for enumerating all inequivalent sets of points (order types) of size 11.
Based on these ndings, we establish a new upper bound on cr(K n )
for general n. The bound stems from a novel construction of drawings
of K n with few crossings.
AMS Subject Classication: 65D18, 05C62, 05A99
Key words: crossing number, complete graph, order types, enumeration
1 Introduction
The crossing number of a graph G is the least number of edge crossings at
tained by a drawing of G in the plane. Drawing may be interpreted in several
ways leading to dierent concepts of crossing number; Pach and Toth [19]
