Editing Four color theorem (section)

===Early proof attempts===
[[File:DeMorganFourColour.png|thumb|Letter of De Morgan to [[William Rowan Hamilton]], 23 Oct. 1852]]
As far as is known,<ref>There is some mathematical folk-lore that [[August Ferdinand Möbius|Möbius]] originated the four-color conjecture, but this notion seems to be erroneous. See {{citation|title=Graph Theory, 1736–1936|title-link=Graph Theory, 1736–1936|author=Biggs, Norman|author-link=Norman L. Biggs|author2=Lloyd, E. Keith|author3=Wilson, Robin J.|author-link3=Robin Wilson (mathematician)|year=1986|publisher=Oxford University Press|isbn=0-19-853916-9|at=[https://books.google.com/books?id=XqYTk0sXmpoC&pg=PA116 p. 116]}} & {{citation|title=Note on the history of the map-coloring problem|author=Maddison, Isabel|author-link=Isabel Maddison|journal=Bull. Amer. Math. Soc.|volume=3|issue=7|year=1897|page=257|doi=10.1090/S0002-9904-1897-00421-9|doi-access=free}}</ref> the conjecture was first proposed on October 23, 1852,<ref name=MacKenzie>Donald MacKenzie, ''Mechanizing Proof: Computing, Risk, and Trust'' (MIT Press, 2004) p103</ref> when [[Francis Guthrie]], while trying to color the map of counties of England, noticed that only four different colors were needed. At the time, Guthrie's brother, [[Frederick Guthrie (scientist)|Frederick]], was a student of [[Augustus De Morgan]] (the former advisor of Francis) at [[University College London]]. Francis inquired with Frederick regarding it, who then took it to De Morgan. (Francis Guthrie graduated later in 1852, and later became a professor of mathematics in South Africa.) According to De Morgan:

<blockquote>A student of mine [Guthrie] asked me to day to give him a reason for a fact which I did not know was a fact—and do not yet. He says that if a figure be any how divided and the compartments differently colored so that figures with any portion of common boundary ''line'' are differently colored—four colors may be wanted but not more—the following is his case in which four colors ''are'' wanted. Query cannot a necessity for five or more be invented...{{sfnp|Wilson|2014|p=12}}</blockquote>

"F.G.", perhaps one of the two Guthries, published the question in ''[[Athenaeum (British magazine)|The Athenaeum]]'' in 1854,<ref>{{harvtxt|F. G.|1854}}; {{harvtxt|McKay|2012}}</ref> and De Morgan posed the question again in the same magazine in 1860.<ref name=Athenaeum1860>{{citation|journal=[[Athenaeum (British magazine)|The Athenaeum]]|date=April 14, 1860|first=Augustus|last=De Morgan (anonymous)|author-link=Augustus De Morgan|pages=501–503|title=The Philosophy of Discovery, Chapters Historical and Critical. By W. Whewell.}}</ref> Another early published reference by {{harvs|first=Arthur|last=Cayley|authorlink=Arthur Cayley|year=1879|txt}} in turn credits the conjecture to De Morgan.

There were several early failed attempts at proving the theorem. De Morgan believed that it followed from a simple fact about four regions, though he didn't believe that fact could be derived from more elementary facts.
<blockquote>This arises in the following way. We never need four colours in a neighborhood unless there be four counties, each of which has boundary lines in common with each of the other three. Such a thing cannot happen with four areas unless one or more of them be inclosed by the rest; and the colour used for the inclosed county is thus set free to go on with. Now this principle, that four areas cannot each have common boundary with all the other three without inclosure, is not, we fully believe, capable of demonstration upon anything more evident and more elementary; it must stand as a postulate.<ref name=Athenaeum1860 /></blockquote>

One proposed proof was given by [[Alfred Kempe]] in 1879, which was widely acclaimed;<ref name="rouse_ball_1960">[[W. W. Rouse Ball]] (1960) ''The Four Color Theorem'', in Mathematical Recreations and Essays, Macmillan, New York, pp 222–232.</ref> another was given by [[Peter Guthrie Tait]] in 1880. It was not until 1890 that Kempe's proof was shown incorrect by [[Percy Heawood]], and in 1891, Tait's proof was shown incorrect by [[Julius Petersen]]—each false proof stood unchallenged for 11 years.{{sfnp|Thomas|1998|p=848}}

In 1890, in addition to exposing the flaw in Kempe's proof, Heawood proved the [[five color theorem]] and generalized the four color conjecture to surfaces of arbitrary genus.{{sfnp|Heawood|1890}}

Tait, in 1880, showed that the four color theorem is equivalent to the statement that a certain type of graph (called a [[snark (graph theory)|snark]] in modern terminology) must be non-[[planar graph|planar]].{{sfnp|Tait|1880}}

In 1943, [[Hugo Hadwiger]] formulated the [[Hadwiger conjecture (graph theory)|Hadwiger conjecture]],{{sfnp|Hadwiger|1943}} a far-reaching generalization of the four-color problem that still remains unsolved.