| | |
Summary: PROGRESS ON RECTILINEAR CROSSING NUMBERS
Oswin Aichholzer
Franz Aurenhammer
Hannes Krasser y
Institute for Theoretical Computer Science
Graz University of Technology
Graz, Austria
e-mail: foaich,auren,hkrasserg@igi.tu-graz.ac.at
Abstract
Let cr(G) denote the rectilinear crossing number of a graph G. We show
cr(K 11
) = 102 and cr(K 12
) = 153. Despite the remarkable hunt for the
crossing number of the complete graph K n , initiated by R. Guy in the 1960s,
these quantities have been unknown for n > 10 to date. We also establish new
upper and lower bounds on cr(K n ) for 13 n 20, along with an improved
general lower bound for cr(K n ). The results mainly rely on recent methods
developed by the authors for exhaustively enumerating all combinatorially
inequivalent sets of points (so-called order types).
1 Introduction and results
|
