In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map...

two adjacent regions receive the same color. The five color theorem is implied by the stronger four color theorem, but is considerably easier to prove...

difficult mathematical problems". In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into...

mathematics, the four-color theorem. They proved that any two-dimensional map, with certain limitations, can be filled in with four colors without any...

theories were developed to reduce the number of colors to four, until the four color theorem was finally proved in 1976 by Kenneth Appel and Wolfgang Haken...

In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference...

Benjamin Werner of INRIA used Coq to create a surveyable proof of the four color theorem, which was completed in 2002. Their work led to the development of...

{\displaystyle 1\leq t\leq 6} . The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging...

prove theorems and to carry out calculations that are too long for any human or team of humans to check; the first proof of the four color theorem is an...

the four color theorem is that every snark is a non-planar graph. Research on snarks originated in Peter G. Tait's work on the four color theorem in 1880...

Grötzsch's theorem is the statement that every triangle-free planar graph can be colored with only three colors. According to the four-color theorem, every...

the American Mathematical Society for his proof with Appel of the four-color theorem. Haken died in Champaign, Illinois, on October 2, 2022, aged 94. Haken's...

Gerda Fritsch to show that Alfred Kempe's attempted proof of the four color theorem was incorrect. The Fritsch graph is one of only six graphs in which...

axiom of choice. Its applications include extending the four-color theorem and Dilworth's theorem from finite graphs and partially ordered sets to infinite...

researchers in the late twentieth century are Fermat's Last Theorem and the four-color theorem. An important open mathematics problem solved in the early...

for deduction: …if we accept the [Four-Color Theorem] as a theorem, we are committed to changing the sense of "theorem", or, more to the point, to changing...

number of colors needed for the plane or sphere, solved in 1976 as the four color theorem by Haken and Appel. On the sphere the lower bound is easy, whereas...

mathematics. He led the formalization of the four color theorem and Feit–Thompson proof of the odd-order theorem. (Both were written using the proof assistant...

letter to the Editor of Nature, Spencer-Brown claimed a proof of the four-color theorem, which is not computer-assisted. The preface of the 1979 edition of...

it in 1921 as a counterexample to Kempe's erroneous proof of the four color theorem; it was named after Errera by Hutchinson & Wagon (1998). The Errera...

in number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants. Computational...

by the four color theorem), and was posed by Gerhard Ringel in 1959. An intuitive form of the problem asks how many colors are needed to color political...

vertices of degree two as well as vertices of higher degree. The four color theorem (proved by Appel & Haken (1976)) on vertex coloring of planar graphs...

method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis...

computational complexity—play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics...

called the "two color theorem"; Soifer credits it to a famous 1879 paper of Alfred Kempe containing a false proof of the four color theorem. Bandelt, H.-J...

significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding...

hdl:2066/75958. S2CID 14827467. Gonthier, Georges (2008), "Formal Proof—The Four-Color Theorem" (PDF), Notices of the American Mathematical Society, 55 (11): 1382–1393...

a certain result, then the theorem is true. A good example of this was the machine-aided proof of the four color theorem, which was very controversial...

"Formal Proof—The Four-Color Theorem" (PDF). Notices of the AMS. 55 (11): 1382. Castelvecchi, Davide (2016-03-01). "Fermat's last theorem earns Andrew Wiles...