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...

Haken at the University of Illinois at Urbana–Champaign, solved the four-color theorem, one of the most famous problems in mathematics. They proved that...

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...

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...

Discharging is most well known for its central role in the proof of the four color theorem. The discharging method is used to prove that every graph in a certain...

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...

projects hope to shorten and simplify this proof. Another theorem of this type is the four color theorem whose computer generated proof is too long for a human...

{\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...

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

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...

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...

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...

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...

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...

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...

Petersen minor, 4-flows exist by the snark theorem (Seymour, et al 1998, not yet published). The four color theorem is equivalent to the statement that no...

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...

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...

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...

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...

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

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...

rhombus, and square. Four is the highest degree general polynomial equation for which there is a solution in radicals. The four-color theorem states that a planar...

Heawood conjecture (now the Ringel–Youngs theorem), a mathematical problem closely linked with the four color theorem. Although born in Austria, Ringel was...

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...