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Example of a four-colored map A four-colored map of the states of the United States (ignoring lakes and oceans). 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 so that no two adjacent regions have the same color.
An entirely different approach was needed for the much older problem of finding the 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 for higher genera the upper bound is easy and was proved in Heawood's original short paper that contained ...
The conjecture was 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 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle ...
It is an extension of the planar map coloring problem (solved by the four color theorem), and was posed by Gerhard Ringel in 1959. [1] An intuitive form of the problem asks how many colors are needed to color political maps of the Earth and Moon, in a hypothetical future where each Earth country has a Moon colony which must be given the same color.
A graph that requires four colors in any coloring, and four connected subgraphs that, when contracted, form a complete graph, illustrating the case k = 4 of Hadwiger's conjecture In graph theory , the Hadwiger conjecture states that if G {\displaystyle G} is loopless and has no K t {\displaystyle K_{t}} minor then its chromatic number satisfies ...
The four color theorem was ultimately proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if ...
Two players, Left and Right, take turns coloring in one uncolored region per turn, subject to various constraints, as in the map-coloring problem. The move constraints and the winning condition are features of the particular game. Some players find it easier to color vertices of the dual graph, as in the Four color theorem. In this method of ...
Foster's theorem ; Four color theorem (graph theory) Four functions theorem (combinatorics) Four-vertex theorem (differential geometry) Fourier inversion theorem (harmonic analysis) Fourier theorem (harmonic analysis) Franel–Landau theorem (number theory) Fraňková–Helly selection theorem (mathematical analysis) Fredholm's theorem (linear ...