Symmetry and the crossing number for complete graphs
Thomas L. Saaty · Journal of Research of the National Bureau of Standards Section B Mathematical Sciences · 1969
This pape r studies the minimum number of intersections of edges in a co mplete graph on n vertices draw n in the plane.The proofs are first given for n ,,;; 10.A theorem on the maximum number of intersections is a lso give n.Geometric representations of these cases are included.Symmetry of the representations is th e n disc ussed as it applies to exte nsions from s mall valu es of n to larger values maintainin g th e minimality of the number of inte rsections.Based on a symmetry co njecture give n in the paper, a proof is give n for the ge neral case of the minimum inte rsection problem .