It does not generalize to surfaces of non-zero genus, though the Heawood conjecture is the analogous statement in those cases, with the exception of the Klein bottle.  
This was solved by the novel method of discharging, which forms the subtitle of the second part of the paper in which its solution was originally published.  It does not generalize to surfaces of non-zero genus, though the Heawood conjecture is the analogous statement in those cases, with the exception of the Klein bottle.  First noted in 1840, it was brought to public attention by the work of de Morgan, who found it infuriating.  FTP, name this problem of graph theory that remained open until a 1977 "computer proof" by Appel and Haken showed that the vertices of "Every Planar Graph" can be uniquely labeled in the namesake way.
This was solved by the novel method of discharging, which forms the subtitle of the second part of the paper in which its solution was originally published.  
four color problem