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Four Colors

Das Festival "Colours of Ostrava" ist zu Ende. Vier Tage Musik auf 8 Bühnen, Theater, Workshops auf einem futuristisch anmutenden Industriegelände in. Bild von DoubleTree by Hilton Hotel Lodz, Lodz: Restauracja Four Colors - Schauen Sie sich 5' authentische Fotos und Videos von DoubleTree by Hilton. Four Colors Suffice: How the Map Problem Was Solved | Wilson, Robin | ISBN: | Kostenloser Versand für alle Bücher mit Versand und Verkauf​.

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This graph is planar : it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves without crossings that lead from one region's vertex, across a shared boundary segment, to an adjacent region's vertex.

Conversely any planar graph can be formed from a map in this way. In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short:.

As far as is known, [6] the conjecture was first proposed on October 23, [7] when Francis Guthrie , while trying to color the map of counties of England, noticed that only four different colors were needed.

Francis inquired with Frederick regarding it, who then took it to De Morgan Francis Guthrie graduated later in , and later became a professor of mathematics in South Africa.

According to De Morgan:. 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…" Wilson , p. 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.

This arises in the following way. We never need four colors 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 color 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.

One alleged proof was given by Alfred Kempe in , which was widely acclaimed; [10] another was given by Peter Guthrie Tait in It was not until that Kempe's proof was shown incorrect by Percy Heawood , and in , Tait's proof was shown incorrect by Julius Petersen —each false proof stood unchallenged for 11 years.

In , 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.

Tait, in , showed that the four color theorem is equivalent to the statement that a certain type of graph called a snark in modern terminology must be non- planar.

In , Hugo Hadwiger formulated the Hadwiger conjecture , [14] a far-reaching generalization of the four-color problem that still remains unsolved.

During the s and s German mathematician Heinrich Heesch developed methods of using computers to search for a proof.

Notably he was the first to use discharging for proving the theorem, which turned out to be important in the unavoidability portion of the subsequent Appel—Haken proof.

He also expanded on the concept of reducibility and, along with Ken Durre, developed a computer test for it. Unfortunately, at this critical juncture, he was unable to procure the necessary supercomputer time to continue his work.

Others took up his methods and his computer-assisted approach. While other teams of mathematicians were racing to complete proofs, Kenneth Appel and Wolfgang Haken at the University of Illinois announced, on June 21, , [16] that they had proved the theorem.

They were assisted in some algorithmic work by John A. If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors.

The proof showed that such a minimal counterexample cannot exist, through the use of two technical concepts: [17]. Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist.

Their proof reduced the infinitude of possible maps to 1, reducible configurations later reduced to 1, which had to be checked one by one by computer and took over a thousand hours.

This reducibility part of the work was independently double checked with different programs and computers. Appel and Haken's announcement was widely reported by the news media around the world, and the math department at the University of Illinois used a postmark stating "Four colors suffice.

In the early s, rumors spread of a flaw in the Appel—Haken proof. In , Appel and Haken were asked by the editor of Mathematical Intelligencer to write an article addressing the rumors of flaws in their proof.

They responded that the rumors were due to a "misinterpretation of [Schmidt's] results" and obliged with a detailed article Wilson , — Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O n 2 time, where n is the number of vertices.

In , Neil Robertson , Daniel P. Sanders , Paul Seymour , and Robin Thomas created a quadratic-time algorithm, improving on a quartic -time algorithm based on Appel and Haken's proof.

Both the unavoidability and reducibility parts of this new proof must be executed by computer and are impractical to check by hand.

In , Benjamin Werner and Georges Gonthier formalized a proof of the theorem inside the Coq proof assistant.

This removed the need to trust the various computer programs used to verify particular cases; it is only necessary to trust the Coq kernel.

Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

The explanation here is reworded in terms of the modern graph theory formulation above. Kempe's argument goes as follows. First, if planar regions separated by the graph are not triangulated , i.

If this triangulated graph is colorable using four colors or fewer, so is the original graph since the same coloring is valid if edges are removed.

So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.

Suppose v , e , and f are the number of vertices, edges, and regions faces. Now, the degree of a vertex is the number of edges abutting it.

If v n is the number of vertices of degree n and D is the maximum degree of any vertex,. If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it four-colorable.

Call this graph G. Kempe also showed correctly that G can have no vertex of degree 4. As before we remove the vertex v and four-color the remaining vertices.

If all four neighbors of v are different colors, say red, green, blue, and yellow in clockwise order, we look for an alternating path of vertices colored red and blue joining the red and blue neighbors.

Such a path is called a Kempe chain. There may be a Kempe chain joining the red and blue neighbors, and there may be a Kempe chain joining the green and yellow neighbors, but not both, since these two paths would necessarily intersect, and the vertex where they intersect cannot be colored.

Suppose it is the red and blue neighbors that are not chained together. Explore all vertices attached to the red neighbor by red-blue alternating paths, and then reverse the colors red and blue on all these vertices.

The result is still a valid four-coloring, and v can now be added back and colored red. This leaves only the case where G has a vertex of degree 5; but Kempe's argument was flawed for this case.

Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument changing only that the minimal counterexample requires 6 colors and use Kempe chains in the degree 5 situation to prove the five color theorem.

In any case, to deal with this degree 5 vertex case requires a more complicated notion than removing a vertex.

Rather the form of the argument is generalized to considering configurations , which are connected subgraphs of G with the degree of each vertex in G specified.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G.

As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

A configuration for which this is possible is called a reducible configuration. If at least one of a set of configurations must occur somewhere in G, that set is called unavoidable.

The argument above began by giving an unavoidable set of five configurations a single vertex with degree 1, a single vertex with degree 2, Because G is triangular, the degree of each vertex in a configuration is known, and all edges internal to the configuration are known, the number of vertices in G adjacent to a given configuration is fixed, and they are joined in a cycle.

These vertices form the ring of the configuration; a configuration with k vertices in its ring is a k -ring configuration, and the configuration together with its ring is called the ringed configuration.

As in the simple cases above, one may enumerate all distinct four-colorings of the ring; any coloring that can be extended without modification to a coloring of the configuration is called initially good.

For example, the single-vertex configuration above with 3 or less neighbors were initially good. In general, the surrounding graph must be systematically recolored to turn the ring's coloring into a good one, as was done in the case above where there were 4 neighbors; for a general configuration with a larger ring, this requires more complex techniques.

Because of the large number of distinct four-colorings of the ring, this is the primary step requiring computer assistance.

Finally, it remains to identify an unavoidable set of configurations amenable to reduction by this procedure.

The primary method used to discover such a set is the method of discharging. The intuitive idea underlying discharging is to consider the planar graph as an electrical network.

Initially positive and negative "electrical charge" is distributed amongst the vertices so that the total is positive.

Each vertex is assigned an initial charge of 6-deg v. Then one "flows" the charge by systematically redistributing the charge from a vertex to its neighboring vertices according to a set of rules, the discharging procedure.

Since charge is preserved, some vertices still have positive charge. The rules restrict the possibilities for configurations of positively charged vertices, so enumerating all such possible configurations gives an unavoidable set.

As long as some member of the unavoidable set is not reducible, the discharging procedure is modified to eliminate it while introducing other configurations.

Appel and Haken's final discharging procedure was extremely complex and, together with a description of the resulting unavoidable configuration set, filled a page volume, but the configurations it generated could be checked mechanically to be reducible.

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Four Colors

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Appel and Haken's announcement was widely reported by the news media around the world, and the math department at the University of Illinois used a postmark stating "Four colors suffice.

In the early s, rumors spread of a flaw in the Appel—Haken proof. In , Appel and Haken were asked by the editor of Mathematical Intelligencer to write an article addressing the rumors of flaws in their proof.

They responded that the rumors were due to a "misinterpretation of [Schmidt's] results" and obliged with a detailed article Wilson , — Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O n 2 time, where n is the number of vertices.

In , Neil Robertson , Daniel P. Sanders , Paul Seymour , and Robin Thomas created a quadratic-time algorithm, improving on a quartic -time algorithm based on Appel and Haken's proof.

Both the unavoidability and reducibility parts of this new proof must be executed by computer and are impractical to check by hand.

In , Benjamin Werner and Georges Gonthier formalized a proof of the theorem inside the Coq proof assistant. This removed the need to trust the various computer programs used to verify particular cases; it is only necessary to trust the Coq kernel.

Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

The explanation here is reworded in terms of the modern graph theory formulation above. Kempe's argument goes as follows.

First, if planar regions separated by the graph are not triangulated , i. If this triangulated graph is colorable using four colors or fewer, so is the original graph since the same coloring is valid if edges are removed.

So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.

Suppose v , e , and f are the number of vertices, edges, and regions faces. Now, the degree of a vertex is the number of edges abutting it. If v n is the number of vertices of degree n and D is the maximum degree of any vertex,.

If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it four-colorable.

Call this graph G. Kempe also showed correctly that G can have no vertex of degree 4. As before we remove the vertex v and four-color the remaining vertices.

If all four neighbors of v are different colors, say red, green, blue, and yellow in clockwise order, we look for an alternating path of vertices colored red and blue joining the red and blue neighbors.

Such a path is called a Kempe chain. There may be a Kempe chain joining the red and blue neighbors, and there may be a Kempe chain joining the green and yellow neighbors, but not both, since these two paths would necessarily intersect, and the vertex where they intersect cannot be colored.

Suppose it is the red and blue neighbors that are not chained together. Explore all vertices attached to the red neighbor by red-blue alternating paths, and then reverse the colors red and blue on all these vertices.

The result is still a valid four-coloring, and v can now be added back and colored red. This leaves only the case where G has a vertex of degree 5; but Kempe's argument was flawed for this case.

Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument changing only that the minimal counterexample requires 6 colors and use Kempe chains in the degree 5 situation to prove the five color theorem.

In any case, to deal with this degree 5 vertex case requires a more complicated notion than removing a vertex. Rather the form of the argument is generalized to considering configurations , which are connected subgraphs of G with the degree of each vertex in G specified.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G.

As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

A configuration for which this is possible is called a reducible configuration. If at least one of a set of configurations must occur somewhere in G, that set is called unavoidable.

The argument above began by giving an unavoidable set of five configurations a single vertex with degree 1, a single vertex with degree 2, Because G is triangular, the degree of each vertex in a configuration is known, and all edges internal to the configuration are known, the number of vertices in G adjacent to a given configuration is fixed, and they are joined in a cycle.

These vertices form the ring of the configuration; a configuration with k vertices in its ring is a k -ring configuration, and the configuration together with its ring is called the ringed configuration.

As in the simple cases above, one may enumerate all distinct four-colorings of the ring; any coloring that can be extended without modification to a coloring of the configuration is called initially good.

For example, the single-vertex configuration above with 3 or less neighbors were initially good. In general, the surrounding graph must be systematically recolored to turn the ring's coloring into a good one, as was done in the case above where there were 4 neighbors; for a general configuration with a larger ring, this requires more complex techniques.

Because of the large number of distinct four-colorings of the ring, this is the primary step requiring computer assistance.

Finally, it remains to identify an unavoidable set of configurations amenable to reduction by this procedure. The primary method used to discover such a set is the method of discharging.

The intuitive idea underlying discharging is to consider the planar graph as an electrical network. Initially positive and negative "electrical charge" is distributed amongst the vertices so that the total is positive.

Each vertex is assigned an initial charge of 6-deg v. Then one "flows" the charge by systematically redistributing the charge from a vertex to its neighboring vertices according to a set of rules, the discharging procedure.

Since charge is preserved, some vertices still have positive charge. The rules restrict the possibilities for configurations of positively charged vertices, so enumerating all such possible configurations gives an unavoidable set.

As long as some member of the unavoidable set is not reducible, the discharging procedure is modified to eliminate it while introducing other configurations.

Appel and Haken's final discharging procedure was extremely complex and, together with a description of the resulting unavoidable configuration set, filled a page volume, but the configurations it generated could be checked mechanically to be reducible.

Verifying the volume describing the unavoidable configuration set itself was done by peer review over a period of several years.

A technical detail not discussed here but required to complete the proof is immersion reducibility. Description: This fun and colorful card game is based on the popular UNO game.

Face up to 3 computer-controlled opponents. Match cards by color or number, play action cards to mix the game up and be the first to get rid of all cards.

Last but not least: Do not forget to press the 1 button when you have only one card left! Category: Card Games.

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  1. Ich meine, dass Sie sich irren. Es ich kann beweisen. Schreiben Sie mir in PM, wir werden besprechen.

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