By revisiting the famous problem — which was controversially solved in the 1970s with the help of computers — mathematicians have gained important new insights into the nature of graphs.

Some math problems continue to haunt researchers long after they’ve been solved. A proof emerges, is even celebrated, and yet dissatisfaction lingers. Perhaps the argument is too convoluted — the hunt persists for the elusive one-page paper — or perhaps it fails to give a deeper theoretical insight into why something is true. Whatever the reason, mathematicians return, again and again, to a case that is otherwise considered closed.

One of the most famous such cases is that of the four-color theorem, a problem that transformed how mathematicians think about their subject.

The problem is simple to state, and even simpler to see: Given a contiguous map, is it possible to color each region with one of four colors such that no neighboring regions share a color? In the mid-19th century, the question was of trifling interest to mapmakers, who had far more than four colors at their disposal and saw no particular reason to restrict their palette. But to mathematicians, both amateur and professional, the brain teaser quickly turned into an obsession.

...read more at quantamagazine.org