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Planar maps require at most five colors
The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the countries of the world,
Five_color_theorem
Planar maps require at most four colors
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
Four_color_theorem
Proposition in mathematics that is unproven
while trying to color the map of counties of England, noticed that only four different colors were needed. The five color theorem, which has a short
Conjecture
Methodic assignment of colors to elements of a graph
However, in that paper he proved the five color theorem, saying that every planar map can be colored with no more than five colors, using ideas of Kempe. In
Graph_coloring
On coloring infinite graphs
axiom of choice. Its applications include extending the four-color theorem and Dilworth's theorem from finite graphs and partially ordered sets to infinite
De Bruijn–Erdős theorem (graph theory)
De_Bruijn–Erdős_theorem_(graph_theory)
Euler's partition theorem (number theory) Fermat polygonal number theorem (number theory) Five color theorem (graph theory) Four color theorem (graph theory)
List_of_theorems
Mathematical device used in proof of the four colour theorem
in the proof of the five color theorem by Percy John Heawood, a weaker but more easily proven version of the four colour theorem. The term "Kempe chain"
Kempe_chain
proof) Erdős–Ko–Rado theorem Euler's formula Euler's four-square identity Euler's theorem Five color theorem Five lemma Fundamental theorem of arithmetic Gauss–Markov
List_of_mathematical_proofs
1998 mathematics book by Aigner and Ziegler
Chapter 39: The five color theorem. Chapter 40: Steve Fisk's proof of the art gallery theorem. Chapter 41: Five proofs of Turán's theorem. Chapter 42: Shannon
Proofs_from_THE_BOOK
Area of discrete mathematics
two regions share a common boundary. This theorem is stronger than five-color theorem. Relatedly, Earth–Moon problem is known for the nowadays open problem
Graph_theory
On coloring the edges of graphs
empty, the theorem trivially holds. Let m > 0 and suppose a proper (Δ+1)-edge-coloring exists for all G − xy where xy ∈ E. We say that color α ∈ {1,..
Vizing's_theorem
In mathematics, a statement that has been proven
mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses
Theorem
Mathematical theorem
The theorem on friends and strangers is a mathematical theorem in an area of mathematics called Ramsey theory. Suppose a party has six people. Consider
Theorem on friends and strangers
Theorem_on_friends_and_strangers
3-regular graph with no 3-edge-coloring
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
Snark_(graph_theory)
Every triangle-free planar graph is 3-colorable
proof from another related theorem: every planar graph with girth at least five is 3-list-colorable. However, Grötzsch's theorem itself does not extend from
Grötzsch's_theorem
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
Errera_graph
American mathematician (1932–2013)
at the University of Illinois at Urbana–Champaign, solved the four-color theorem, one of the most famous problems in mathematics. They proved that any
Kenneth_Appel
Sufficiency theorem for reconstructing signals from samples
The Nyquist–Shannon sampling theorem, or the sampling theorem, is a theorem in the field of signal processing which serves as a fundamental bridge between
Nyquist–Shannon sampling theorem
Nyquist–Shannon_sampling_theorem
Theorem in projective geometry
In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem, Latin for mystical hexagram) states that if six arbitrary points
Pascal's_theorem
British mathematician (1861–1955)
daughter. Heawood conjecture Heawood number Heawood graph Four color theorem Five color theorem "The Saving of Durham Castle". Times obituary. Retrieved 26
Percy_John_Heawood
weaker five color theorem. The four-color theorem was eventually proved by Kenneth Appel and Wolfgang Haken in 1976. Schröder–Bernstein theorem. In 1896
List_of_incomplete_proofs
Statement in mathematical combinatorics
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours)
Ramsey's_theorem
Complements of perfect graphs are perfect
In graph theory, the perfect graph theorem of László Lovász (1972a, 1972b) states that an undirected graph is perfect if and only if its complement graph
Perfect_graph_theorem
1990 book by Gerhard Ringel and Nora Hartsfield
Cayley's formula; graph labelings; planar graphs, the four color theorem, and the circle packing theorem; near-planar graphs; and graph embedding on topological
Pearls_in_Graph_Theory
Unproven generalization of the four-color theorem
{\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 open
Hadwiger conjecture (graph theory)
Hadwiger_conjecture_(graph_theory)
Existence of a line through two points
The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the
Sylvester–Gallai_theorem
Mathematical problem
bounded by Jordan curves, then at least six colors are required. Four color theorem Hadwiger conjecture Soifer (2008), pp. 557–563; Shelah & Soifer (2003)
Hadwiger–Nelson_problem
Graph coloring with equal color classes
maximum degree five, the number of colors guaranteed for it by the Hajnal–Szemerédi theorem is six, achieved by giving each vertex a distinct color. Another
Equitable_coloring
Graph that can be embedded in the plane
{\displaystyle 30.06^{n}} . The four color theorem states that every planar graph is 4-colorable (i.e., 4-partite). Fáry's theorem states that every simple planar
Planar_graph
Figure that, added to a given figure, makes a larger figure of the same shape
{\begin{matrix}\color {red}1&\color {orange}2&\color {yellow}3&\color {green}4&\color {blue}5&\color {indigo}6&\color {violet}7&\color {pink}8\\\color {orange}2&\color
Gnomon_(figure)
Geometry theorem
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that if A , B , C {\displaystyle A,B,C} is one set of collinear points
Pappus's_hexagon_theorem
Mathematical proof at least partially generated by computer
of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
Computer-assisted_proof
Result about flat-foldable origami crease patterns
Maekawa's theorem is a theorem in the mathematics of paper folding named after Jun Maekawa. It relates to flat-foldable origami crease patterns and states
Maekawa's_theorem
Concept in graph theory
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
Nowhere-zero_flow
Graph divided into two independent sets
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
Bipartite_graph
Algorithm for division of polynomials
For example, if the rational root theorem produces a single (rational) root of a quintic polynomial (degree five), it can be factored out to obtain a
Polynomial_long_division
German mathematician (1910–2000)
D. in 1937, with a dissertation concerning the Jordan curve theorem and four color theorem, and taught at Cologne for many years himself. In 1970, he moved
Klaus_Wagner
4-color theorem. Appel and Haken's proof of this took 139 pages, and also depended on long computer calculations. 1974 – The Gorenstein–Harada theorem classifying
List of long mathematical proofs
List_of_long_mathematical_proofs
Assignment of colors to graph vertices that destroys all symmetries
automorphisms, with distinguishing number two. This result extends Frucht's theorem that every finite group can be realized as the group of symmetries of a
Distinguishing_coloring
Assignment of colors to edges of a graph
called the chromatic index of the graph. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree Δ or Δ+1
Edge_coloring
Formal argument for the existence of God
"possibly exemplified", i.e. applies at least to some object in some world (theorem 1). Defining an object to be Godlike if it has all positive properties
Gödel's_ontological_proof
Upper bound on a graph's Shannon capacity
28. Lovász (1979), Theorem 3. Lovász (1979), Theorem 4. Lovász (1979), Theorem 5. Riddle (2003). Lovász (1979), Lemma 2 and Theorem 7. Lovász (1979), Corollary
Lovász_number
Type of mathematical expression
color {Red}{P}}{\color {Blue}{Q}}&{=}&&({\color {Red}{2x}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{2x}}\cdot
Polynomial
Branch of discrete mathematics
none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory. The types of questions
Combinatorics
Branch of mathematical logic
are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast
Reverse_mathematics
Convex polyhedron with 14 triangle faces
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
Triaugmented_triangular_prism
Private, all-girls school in New Orleans, Louisiana, United States
Jackson, Ne'Kiya; Johnson, Calcea (October 27, 2024). "Five or Ten New Proofs of the Pythagorean Theorem". The American Mathematical Monthly. 131 (9): 739–752
St. Mary's Academy (New Orleans)
St._Mary's_Academy_(New_Orleans)
Type of permutation
}&{\color {blue}2}&{\color {blue}4}&{\color {blue}5}&{\color {red}5}\\{\color {red}16}&{\color {blue}16}&{\color {blue}14}&{\color {blue}10}&{\color {blue}5}&{\leftarrow
Alternating_permutation
Unit-distance-preserving maps are isometries
In geometry, the Beckman–Quarles theorem states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit
Beckman–Quarles_theorem
Persian mathematician (1265–1318)
theorem of arithmetic. Asas al-qawa'id fi usul al-fawa'id (The base of the rules in the principles of uses) which comprises an introduction and five chapters
Kamāl_al-Dīn_al-Fārisī
Five-pointed star polygon
pentagram Petersen graph – Cubic graph with 10 vertices and 15 edges Ptolemy's theorem – Relates the 4 sides and 2 diagonals of a quadrilateral with vertices
Pentagram
Gluing graphs at complete subgraphs
the eight-vertex Wagner graph; this structure theorem can be used to show that the four color theorem is equivalent to the case k = 5 of the Hadwiger
Clique-sum
Abstract strategy board game
their color on an empty intersection. Black plays first. The winner is the first player to form an unbroken line of five stones of their color horizontally
Gomoku
Swiss mathematician (1908–1981)
any cover of the plane by five congruent closed sets contains a unit distance in one of the sets. Hadwiger proved a theorem characterizing eutactic stars
Hugo_Hadwiger
On complete subdivisions in nonplanar graphs
conjecture appeared in Journal of Combinatorial Theory, Series B. Four-color theorem Hajós' conjecture Condie, Bill (May 30, 2016), "Maths mystery solved
Kelmans–Seymour_conjecture
Graph with tight clique-coloring relation
important minimax theorems in combinatorics, including Dilworth's theorem and Mirsky's theorem on partially ordered sets, Kőnig's theorem on matchings, and
Perfect_graph
First article on transfinite set theory
Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary
Cantor's first set theory article
Cantor's_first_set_theory_article
Abstract strategy board game whose pieces are polyominoes
App Store in December 2018. Family Games: The 100 Best Domino Four color theorem Pentomino Polyomino Pseudo-polyomino Tetromino Tromino Tetris Goodridge
Blokus
Theorem in combinatorics
choice Blichfeldt's theorem Combinatorial principles Combinatorial proof Dedekind-infinite set Dirichlet's approximation theorem Hilbert's paradox of
Pigeonhole_principle
Positive real number which when multiplied by itself gives 5
of the rational numbers. The Kronecker–Weber theorem therefore guarantees that the square root of five can be written as a rational linear combination
Square_root_of_5
Non-abelian group of order eight
Brown 1982, p. 101, exercise 1 Cartan & Eilenberg 1999, Theorem 11.6, p. 262 Brown 1982, Theorem 4.3, p. 99 Roman, Steven (2011). Fundamentals of Group
Quaternion_group
Field of knowledge
and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is widely used to model and solve
Mathematics
Greek philosopher (c. 570 – c. 495 BC)
mathematical and scientific discoveries, such as the Pythagorean theorem, Pythagorean tuning, the five regular solids, the theory of proportions, the sphericity
Pythagoras
Cubic graph with 10 vertices and 15 edges
that no edge connects vertices of the same color. It has a list coloring with 3 colors, by Brooks's theorem for list colorings. The Petersen graph has
Petersen_graph
Shape with three equal sides
the Reuleaux triangle Many theorems, inequalities, and problems involves equilateral triangles. For example, Napoleon's theorem stated that the centroid
Equilateral_triangle
Subatomic particle having no substructure
seventeen distinct particles—twelve fermions and five bosons. As a consequence of flavor and color combinations and antimatter, the fermions and bosons
Elementary_particle
48 kHz, a common sampling rate
double the maximal frequency carried (as per the Nyquist–Shannon sampling theorem) — at least 40 kHz to roughly cover all human-audible frequencies without
48,000_Hz
Indian mathematician (c. 598–c. 668)
Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle's base is divided into
Brahmagupta
Size of largest complete graph made by contracting edges of a given graph
characterization of the graphs with this Hadwiger number) to the four color theorem on colorings of planar graphs, and the conjecture has also been proven
Hadwiger_number
Seven mathematical problems with a US$1 million prize for each solution
Paul Wolfskehl (offered a cash prize for the solution to Fermat's Last Theorem) Smale's problems "Последнее "нет" доктора Перельмана". Interfax. July
Millennium_Prize_Problems
Triangle-free graph requiring four colors
used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable. The Grötzsch graph is a member of an infinite
Grötzsch_graph
Aesthetic value of mathematics
For example, Kenneth Appel and Wolfgang Haken's proof of the four color theorem made use of computer checking of over a thousand cases. Philip J. Davis
Mathematical_beauty
Paper-and-pencil game for two players
of either color. They must alternate colors after each successful landing and must be careful not to block themself. Hales–Jewett theorem m,n,k-game
Tic-tac-toe
"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
History_of_mathematics
Austrian physicist (1887–1961)
physics: statistical mechanics and thermodynamics, physics of dielectrics, color theory, electrodynamics, general relativity, and cosmology, and he made
Erwin_Schrödinger
Conjecture in graph theory
Ramsey's theorem proves that no graph has both its maximum clique size and maximum independent set size smaller than logarithmic. Ramsey's theorem also implies
Erdős–Hajnal_conjecture
Branch of pure mathematics
understand but are very difficult to solve. Examples of this are Fermat's Last Theorem, which was proved 358 years after the original formulation, and Goldbach's
Number_theory
Polyhedron that tiles space by translation
same normal vectors and face areas, contradicting Minkowski's uniqueness theorem. Each face of a parallelohedron must also be centrally symmetric, to match
Parallelohedron
Persian astronomer (1201–1274)
mathematicians do not normally claim work like other scientists, so declaring a theorem for oneself is an exception and not the norm. Therefore, there is motive
Nasir_al-Din_al-Tusi
root of two correctly to five decimal places, and contains "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it
Timeline_of_mathematics
Painting by Jean Metzinger
mathematicians; we should have to study, at some length, certain of Riemann's theorems." The concept of observing a subject from different points in space and
Woman_with_a_Horse
British record producer and keyboardist (born 1986)
artists he has collaborated with, including S10. In 2023, Fish engineered "Theorem" by Puscifer from their album Existential Reckoning: Re-Wired. The same
Jordan_Fish
Hypothetical physical concept
Gödel's incompleteness theorem suggests that attempts to construct a theory of everything are bound to fail. Gödel's theorem, informally stated, asserts
Theory_of_everything
the color red, used in capes by professional bullfighters. Cattle are dichromats, so red does not stand out as a bright color. It is not the color of the
List of common misconceptions about science, technology, and mathematics
List_of_common_misconceptions_about_science,_technology,_and_mathematics
Shape with four equal sides and angles
number of equal-area triangles, a result of Monsky's theorem. Cross's theorem or Vecten's theorem states that, for a triangle formed by the sides of three
Square
Canadian-American mathematician (born 1938)
Seymour, Thomas, and Daniel P. Sanders published a new proof of the four color theorem, confirming the Appel–Haken proof which until then had been disputed
Neil Robertson (mathematician)
Neil_Robertson_(mathematician)
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
particularly tractable representation theory because of the Peter–Weyl theorem. Just like simple complex Lie algebras, centerless compact Lie groups are
Simple_Lie_group
Natural number
domain that is a regular hyperbolic 20-sided icosagon. By the Gauss-Bonnet theorem, the area of this fundamental polygon is equal to 12 π {\displaystyle 12\pi
12_(number)
Special mathematical function
{cornflowerblue}2701}+{\color {navy}15}\times {\color {cornflowerblue}184}+{\color {navy}150}\times {\color {cornflowerblue}13}+{\color {navy}1707}\times {\color {cornflowerblue}1}{\bigr
Nome_(mathematics)
Graph coloring with an allowed number of same-color neighbors
are not (3, 1)-colorable, but every planar graph is (3, 2)-colorable. Together with the (4, 0)-coloring implied by the four color theorem, this solves defective
Defective_coloring
Mathematical tree with cycle through leaves
removed, the remaining graph will no longer be 3-connected. By Steinitz's theorem, as a 3-connected planar graph, it can be represented as the set of vertices
Halin_graph
Group of 14th-century English mathematicians and philosophers
speed theorem (though it was later credited to Galileo) which is also known as "The Law of Falling Bodies". A basic definition of the mean speed theorem is;
Oxford_Calculators
theory. 2. A color class of a colored graph is the set of vertices or edges having one particular color. 3. In the context of Vizing's theorem, on edge coloring
Glossary_of_graph_theory
Two tetrahedra joined by one face
otherwise, the triangular bipyramid is oblique. According to Steinitz's theorem, a graph can be represented as the skeleton of a polyhedron if it is a
Triangular_bipyramid
Board game
sub-boards (or quadrants). Taking turns, the two players place a marble of their color (either black or white) onto an unoccupied space on the board, and then
Pentago
Size of a set in mathematics
are proven to be uncountable by so-called diagonal arguments. Cantor's theorem generalizes these arguments to show there is an infinite hierarchy of infinities
Cardinality
Graph coloring where each vertex has a list of allowed colors
meaning no two adjacent vertices receive the same color. A graph is k-choosable (or k-list-colorable) if it has a proper list coloring no matter how one
List_coloring
Measure of the state of democracy according to The Economist
entrenched and the latter struggling to democratize. The threshold for each color has also been changed from greater than the integer to greater than or equal
The_Economist_Democracy_Index
Hungarian and American mathematician and physicist (1903–1957)
the application of this work was instrumental in his mean ergodic theorem. The theorem is about arbitrary one-parameter unitary groups t → V t {\displaystyle
John_von_Neumann
Ball used in cue sports
such as carom billiards, pool, and snooker. The number, type, diameter, color, and pattern of the balls differ depending upon the specific game being
Billiard_ball
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FIVE COLOR-THEOREM
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FIVE COLOR-THEOREM
FIVE COLOR-THEOREM
FIVE COLOR-THEOREM
FIVE COLOR-THEOREM
FIVE COLOR-THEOREM
FIVE COLOR-THEOREM
FIVE COLOR-THEOREM
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