Map Coloring (Wolfram)

    <p class=default>
      
        When the four-color theorem was proven, the Heawood formula was shown to hold also for all 
        orientable and nonorientable surfaces with the exception of the Klein bottle. For this 
        case, the actual number of colors N needed is six--one less than  (Franklin 1934; Saaty 
        1986, p. 45).
      
    </p>

<p>

      
        surface 	g 	 	N
      
    </p>

<p>

      
        Klein bottle 	1 	7 	6
      
    <br/>

      
        Möbius strip 		6 	6
      
    <br/>

      
        plane 	0 	4 	4
      
    <br/>

      
        projective plane 		6 	6
      
    <br/>

      
        sphere 	0 	4 	4
      
    <br/>

      
        torus 	1 	7 	7
      
    </p>