 
Summary: On ( k)pseudoedges in generalized configurations
and the pseudolinear crossing number of Kn
B. ŽAbrego
J. Balogh
S. FernŽandezMerchant
J. Lea~nos
G. Salazar
July 18, 2006
Abstract
It is known that every generalized configuration with n points has at least 3
`k+2
2
Ž
( k)pseudoedges,
and that this bound is tight for k n/3  1. Here we show that this bound is no longer tight for (any)
k > n/3  1. As a corollary, we prove that the usual and the pseudolinear (and hence the rectilinear)
crossing numbers of the complete graph Kn are different for every n 10. It has been noted that
all known optimal rectilinear drawings of Kn share a triangularlike property, which we abstract into
the concept of 3decomposability. We give a lower bound for the crossing numbers of all pseudolinear
drawings of Kn that satisfy this property. This bound coincides with the best general lower bound known
