Editing Three utilities problem (section)

===Generalizations===
[[File:K33 one crossing.svg|thumb|upright=0.5|Drawing of <math>K_{3,3}</math> with one crossing]]
Two important characterizations of planar graphs, [[Kuratowski's theorem]] that the planar graphs are exactly the graphs that contain neither <math>K_{3,3}</math> nor the [[complete graph]] <math>K_5</math> as a subdivision, and [[Wagner's theorem]] that the planar graphs are exactly the graphs that contain neither <math>K_{3,3}</math> nor <math>K_5</math> as a [[minor (graph theory)|minor]], make use of and generalize the non-planarity of <math>K_{3,3}</math>.{{r|little}}

[[Pál Turán]]'s "[[Turán's brick factory problem|brick factory problem]]" asks more generally for a formula for the [[crossing number (graph theory)|minimum number of crossings]] in a drawing of the [[complete bipartite graph]] <math>K_{a,b}</math> in terms of the numbers of vertices <math>a</math> and <math>b</math> on the two sides of the bipartition. The utility graph <math>K_{3,3}</math> may be drawn with only one crossing, but not with zero crossings, so its crossing number is one.{{r|early|ps09}}{{Clear|left}}