K. Appel and W. Haken. Every planar map is four colorable. Illinois J. Math, 21:429--567, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with four or fewer colors (i.e. labels) so that no two adjacent regions get the same color; two regions are adjacent if they share a common boundary curve. This famous theorem, first conjectured in 1852 and proved at last in 1976, would seem to already have a solution in the sense of a proof [4], 36] However, for reasons outlined below, the existing results are not fully satisfactory. The Four Color Theorem appears to be a statement about the topological properties of the plane but on deeper study it reveals aspects that are of different dimensionality. While seemingly about the ....

....governs the phenomena they study. This would be important since it is known that quantum computation based on a nonlinear model can achieve polynomial solutions to NP complete problems. 5 Coloring as a test problem The four color theorem was originally proved by Appel, Haken and Koch [2] 3] [4] using a combination of traditional mathematics and computer implemented calculations. Though their work depended on some heuristic innovations by Heesch [11] its broad outline is basically as suggested by Birkhoff [7] 60 years earlier. A set of unavoidable configurations is generated so that ....

K. Appel and W. Haken, Every planar map is four colorable, Contemporary Math. 98 (1989) entire issue.

....by Delta. Theorem 1 Let A be a particular complex number and let I be the Jones Wenzl idempotent element in the Temperley Lieb algebra with parameter A. Then Delta(I = n 1] q ; where q = A . Thus, there is an interesting possible choice for integer representation. For example, [3] A = 1 (A ) e 1 ; where as above denotes addition in the algebra and is scalar multiplication. 7 I believe that one could use unitary evolution operators to produce the required qubit topologies. However, any implementation of quantum computing via such abstract ....

....governs the phenomena they study. This would be important since it is known that quantum computation based on a nonlinear model can achieve polynomial solutions to NP complete problems. 5 Coloring as a test problem The four color theorem was originally proved by Appel, Haken and Koch [2] [3], 4] using a combination of traditional mathematics and computer implemented calculations. Though their work depended on some heuristic innovations by Heesch [11] its broad outline is basically as suggested by Birkhoff [7] 60 years earlier. A set of unavoidable configurations is generated so ....

K. Appel, W. Haken and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J. Math. 21 (1977), 491--567.

....However, their products are not mirror symmetric in this way. The Temperley Lieb algebra is generated by these elements and their products and complex linear combinations, subject to a family of relations: e i e j = e j e i when ki Gamma jk 2, e i 1 e i e i 1 = e i e i 1 e i and e i = [2] q e i , where q = A and [2] q = q q . See, e.g. 35] The first reference we could find to the idea of a linkage between the areas of quantum algebra and quantum computing appears in Fivel [8] which gives a connection between the algebraic geometry methods of classical encryption and a ....

....are not mirror symmetric in this way. The Temperley Lieb algebra is generated by these elements and their products and complex linear combinations, subject to a family of relations: e i e j = e j e i when ki Gamma jk 2, e i 1 e i e i 1 = e i e i 1 e i and e i = 2] q e i , where q = A and [2] q = q q . See, e.g. 35] The first reference we could find to the idea of a linkage between the areas of quantum algebra and quantum computing appears in Fivel [8] which gives a connection between the algebraic geometry methods of classical encryption and a physical aspect of quantum ....

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K. Appel and W. Haken, Every planar map is four colorable. Part I. Discharging, Illinois J. Math. 21 (1977), 429-490.

....colors so that any two adjacent vertices get di erent colors. The chromatic number of G, denoted by (G) is the smallest k such that G is k colorable. We say that G is k chromatic if (G) k. It is well known that every planar graph G has chromatic number at most 4, the Four Color Theorem [1, 22]. However, if we restrict G to be a quadrangulation, then the chromatic number decreases to 2 since every quadrangulation on the sphere is bipartite. For general closed surfaces F , the chromatic number of a graph G which is embedded in can be bounded by: 6 6 7 7 7 This is best ....

K. Appel and W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976) 449-456.

....proof of the four color theorem. However, such a research program has yet to be carried out to success. In more recent times the activity directed towards this goal has been somewhat reduced following the combinatorial proof of the four color theorem by K. Appel, W. Haken and J. Koch in 1976 ([1]) Since it is, from experience, extremely dicult to calculate the chromatic polynomial of a graph, there is not much promise of obtaining a practical method along these lines for deciding about colorability of any given planar graph, or of any given general graph for that matter. The objective ....

K. Appel, W. Haken and J. Koch, Every planar map is four colorable, Part I and II, Illinois J. Math. 21 (1977) 429-567.

....all together, an even number of edges. THEOREM 19.1.10 Barnette [10] 1966) Every 3 polytopal graph contains a spanning tree of maximal valence 3. We will now describe some results and a conjecture on colorability and Hamiltonian circuits. THEOREM 19.1. 11 Four Color Theorem: Appel and Haken [4, 5, 92] (1977) The graph of every 3 polytope is 4 colorable. 4 connected planar graphs are Hamiltonian. Tait conjectured in 1880, and Tutte disproved in 1946, that the graph of every simple 3 polytope is Hamiltonian. This started a rich theory of trivalent planar graphs without large paths. Every ....

K. Appel and W. Haken. Every Planar Map is Four Colorable, volume 98 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1989.

....all together, an even number of edges. THEOREM 19.1.10 Barnette [10] 1966) Every 3 polytopal graph contains a spanning tree of maximal valence 3. We will now describe some results and a conjecture on colorability and Hamiltonian circuits. THEOREM 19.1. 11 Four Color Theorem: Appel and Haken [4, 5, 92] (1977) The graph of every 3 polytope is 4 colorable. 4 connected planar graphs are Hamiltonian. Tait conjectured in 1880, and Tutte disproved in 1946, that the graph of every simple 3 polytope is Hamiltonian. This started a rich theory of trivalent planar graphs without large paths. Every ....

K. Appel and W. Haken. Every planar map is four colorable. American Mathematical Society Bulletin, 82(5):711-712, 1976.

....were originally made with the belief that they were solutions. These include work by Kempe and Tait. Indeed, the 4 Color Problem motivated many of the now central problems in graph theory. For details see [4] The problem was nally solved (yes, 4 colors do suce ) in 1977 by Appel and Haken [1, 2]. Their proof was at rst controversial, in part because 9 of their use of long computer computations. Since then these calculations have been done independently, including a very careful check by Robertson, Sanders, Seymour, and Thomas [15] We close this section with a variation on the 4 Color ....

K. Appel and W. Haken, Every planar map is four colorable, Part II: Reducibility, Illinois J. Math. 21 (1977) 491-567.

....were originally made with the belief that they were solutions. These include work by Kempe and Tait. Indeed, the 4 Color Problem motivated many of the now central problems in graph theory. For details see [4] The problem was nally solved (yes, 4 colors do suce ) in 1977 by Appel and Haken [1, 2]. Their proof was at rst controversial, in part because 9 of their use of long computer computations. Since then these calculations have been done independently, including a very careful check by Robertson, Sanders, Seymour, and Thomas [15] We close this section with a variation on the 4 Color ....

K. Appel and W. Haken, Every planar map is four colorable, Part I: Discharging, Illinois J. Math. 21 (1977) 429-490.

....mathematics. Fully automated proofs in graph theory are still limited to simple properties [50] 51] 52] 56] 64] 65] 66] In contrast, partly automated proofs, which use both human reasoning and specialized computer programs, have met with much success since the proof of the 4 color theorem [2] 3][4][116] and despite the controversy on the reliability of such proofs, see e.g. 11] To illustrate, the fth update of a dynamic survey on Small Ramsey Numbers [112] reviews results which were obtained with the aid of the computer in 71 papers among the 274 which are cited. Recourse to the ....

Appel, K., and Haken, W. Every planar map is four colorable. Contemp. Math., 98 (1989), 1 - 741.

....of mathematics. Fully automated proofs in graph theory are still limited to simple properties [50] 51] 52] 56] 64] 65] 66] In contrast, partly automated proofs, which use both human reasoning and specialized computer programs, have met with much success since the proof of the 4 color theorem [2][3][4] 116] and despite the controversy on the reliability of such proofs, see e.g. 11] To illustrate, the fth update of a dynamic survey on Small Ramsey Numbers [112] reviews results which were obtained with the aid of the computer in 71 papers among the 274 which are cited. Recourse to the ....

Appel, K., and Haken, W. Every planar map is four colorable. Part II. Reducibility. Illinois J. Math., 21 (1977), 491 - 567.

....elds of mathematics. Fully automated proofs in graph theory are still limited to simple properties [50] 51] 52] 56] 64] 65] 66] In contrast, partly automated proofs, which use both human reasoning and specialized computer programs, have met with much success since the proof of the 4 color theorem [2][3] 4] 116] and despite the controversy on the reliability of such proofs, see e.g. 11] To illustrate, the fth update of a dynamic survey on Small Ramsey Numbers [112] reviews results which were obtained with the aid of the computer in 71 papers among the 274 which are cited. Recourse to ....

Appel, K., and Haken, W. Every planar map is four colorable. Part I. Discharging. Illinois J. Math., 21 (1977), 429 - 490.

....Morgan believed the 4 colour theorem to be true, though neither could prove it. Kempe provided an incorrect 5 colour theorem in 1880, corrected ten years later by Heawood. However, it was not until 1976 that Appel, Haken and Koch were able to deliver a proof of the 4 colour theorem for all cases [AH77, AHK77], and not until 1996 that Robertson, Sanders, Guthrie actually posed the question in terms of a map of the world, though the analogy that countries may be represented by vertices, and their borders similarly by edges, provides the obvious link. Figure 2.2: K 3;3 and K 5 , Kuratowski s ....

K. Appel, W. Haken, and J. Koch. Every planar map is four colorable. Part II: Reducibility. Illinois J. Math., 21:491--567, 1977. 12

....Morgan believed the 4 colour theorem to be true, though neither could prove it. Kempe provided an incorrect 5 colour theorem in 1880, corrected ten years later by Heawood. However, it was not until 1976 that Appel, Haken and Koch were able to deliver a proof of the 4 colour theorem for all cases [AH77, AHK77], and not until 1996 that Robertson, Sanders, Guthrie actually posed the question in terms of a map of the world, though the analogy that countries may be represented by vertices, and their borders similarly by edges, provides the obvious link. Figure 2.2: K 3;3 and K 5 , Kuratowski s ....

K. Appel and W. Haken. Every planar map is four colorable. Part I: Discharging. Illinois J. Math., 21:429--490, 1977. 12

....proofs for 11 some problems that have so far eluded the search for simple proofs. The most famous example of a theorem that can (so far) only be proven with the help of a computer is the Four Colour Theorem, which states that four colours suffice to colour a two dimensional map of countries [29, 30]. Another example is the proof of the Robbins conjecture by an automated theorem prover in 1996 [31] A boolean algebra is a set with an unary negation operator # and and a binary operator # fulfilling associativity and commutativity as well as ###x # y# ####x ##y# # x. The Robbins conjecture ....

K. Appel, W. Haken and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J. Math. 21 (1977) 491--567.

....proofs for 11 some problems that have so far eluded the search for simple proofs. The most famous example of a theorem that can (so far) only be proven with the help of a computer is the Four Colour Theorem, which states that four colours suffice to colour a two dimensional map of countries [29, 30]. Another example is the proof of the Robbins conjecture by an automated theorem prover in 1996 [31] A boolean algebra is a set with an unary negation operator # and and a binary operator # fulfilling associativity and commutativity as well as ###x # y# ####x ##y# # x. The Robbins conjecture ....

K. Appel and W. Haken, Every planar map is four colorable. Part I. Discharging, Illinois J. Math. 21 (1977) 429--490. 66

....The present paper is a contribution to this area. We consider (unless otherwise stated) oriented graphs, that is directed graphs not containing two opposite arcs. Oriented graphs are thus orientations of undirected graphs. Motivated by 3 , 4 and 5 color theorems for (undirected) planar graphs [1, 6, 20], we study the similar questions for oriented graphs. It appears that these questions remain interesting even under of the large girth assumptions in the range where the chromatic number is an easy invariant. This is the main motivation of this paper. More precisely, we consider the following ....

K. Appel and W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976), 711--712.

....say that y forbids k colors for x if for a given choice of the color of y we still have in any case n Gamma k possible choices for coloring x. All the proofs will be based on the so called method of reducible configurations, contributed to Heesh [5] and used in particular by Appel and Haken [1] in their proof of the four color theorem. We first provide a (small) set of forbidden configurations, that is a set of graphs that a minimal counter example G to our theorem cannot contain as subgraphs. We will then assume that every vertex v in G is valued by it degree deg(v) and define a ....

K. Appel and W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976), 711--712.

....numerous times in computer science and mathematics the last few decades. Several papers on approximation algorithms present results based on numerical evidence, and there are several proofs in mathematics in which computers play central roles, e.g. Appel and Haken s proof of the four color theorem [8]. In spite of all the results above being based on calculations which would be too cumbersome to perform by hand in reasonable time, we feel that the degrees to which they can be trusted di#er. This is because it for some calculations is possible to verify the numerical results by hand, while for ....

Kenneth Appel and Wolfgang Haken. Every planar map is four colorable. Illinois Journal of Mathematics, 21:429#567, 1977. 125

....the use of efficient decision procedures, proved correct once and for all, to replace long proof derivations. A typical example is the use of computations on binary decision diagrams to prove results on boolean formulas [23] Another example is Appel and Haken s famous proof of the 4 color theorem [2], which involves checking colorability of a finite, but large, number of well chosen planar graphs: rather than developing a proof of 4 colorability for every such graph, a proved decision procedure is invoked on all of them. The present paper reports on the design and implementation of a strong ....

K. Appel and W. Haken. Every planar map is four colorable. Illinois J. Math, 21:429--567, 1977.

....graphs contain triangles which cover all edges of the graph. For these graphs we can save one additional color as shown below. Theorem 2.2 Let G be a planar graph with at least one edge such that each edge of G is contained in some 3 cycle of G. Then #(H(G) 2. Proof. By the 4 Color Theorem [1, 7], there is a 4 coloring of G. For i = 1, 2, 3, 4, let U i #V (G) be the set of vertices colored i. Now, let c(v) 1 if v 2 , and let c(v) 2 if v 4 . Since every maximal clique K in G contains at least 3 vertices, K uses at least 3 colors in the 4 coloring of G, and hence c uses both ....

K. Appel and W. Haken, Every planar map is four colorable, Contemporary Mathematics 98, AMS, 1989.

....has no loop edges. A graph is planar if it can be drawn in the plane without crossings. A 4 coloring of a graph G is a function c: V(G) 1,2,3,4 such that c(u) c(v) whenever u, v are the ends of some edge of G. A graph is 4 colorable if it admits at least one 4 coloring. The Four Color Theorem [1, 2, 8] asserts the following. 2.1) Every planar graph is 4 colorable if and only if it is loopless. The only if part is, of course, trivial every 4 colorable (not necessarily planar) graph is loopless, but the converse is much much harder. So far, there are only two proofs of (2.1) and both are ....

K. Appel, W. Haken and J. Koch, Every planar map is four colorable, Part II: re- ducibility, Illinois J. of Math. 21 (1977), 491-567.

....has no loop edges. A graph is planar if it can be drawn in the plane without crossings. A 4 coloring of a graph G is a function c: V(G) 1,2,3,4 such that c(u) c(v) whenever u, v are the ends of some edge of G. A graph is 4 colorable if it admits at least one 4 coloring. The Four Color Theorem [1, 2, 8] asserts the following. 2.1) Every planar graph is 4 colorable if and only if it is loopless. The only if part is, of course, trivial every 4 colorable (not necessarily planar) graph is loopless, but the converse is much much harder. So far, there are only two proofs of (2.1) and both are ....

K. Appel and W. Haken, Every planar map is four colorable, Part I: discharging, Illinois J. of Math. 21 (1977), 429-490.

....can be 5 colored provided all noncontractible cycles have length at least 2 (14g 5) This 1 2 K. L. COLLINS AND J. P. HUTCHINSON result was rst shown for the torus in [3] Hence these large edge width graphs do not di er greatly in chromatic number from planar graphs, which can be 4 colored [5, 20]. Another example of a planar k coloring theorem with a companion (k 1) coloring theorem for the same class of graphs embedded on surfaces, but with large edge width, is the following. Call graphs that can be embedded in a surface with each region bounded by an even number of edges evenly ....

K. Appel and W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc. 82 (1976) 711-712.

....Problems Many problems and questions have been raised by our study of iterated colorings of graphs. We conclude by providing a list of some of the most interesting ones. 1. What can you say about # # (G) and ir # (G) for planar graphs Since i # (G) #(G) we know from the Four Color Theorem [1, 2] that if G is planar, then ir # (G) # # # (G) # i # (G) #(G) # 4. Can you prove that, for planar graphs G, # # (G) # 4, without using the Four Color Theorem Failing this, can you prove that ir # (G) # 4, for planar graphs G, without using the Four Color Theorem 2. Investigate ....

K. Appel, W. Haken, and J. Koch, Every planar map is four colorable. Part II: Reducibility, Illinois J. Math. 21:491-567. 1977.

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K. Appel and W. Haken. Every planar map is four colorable. Illinois J. Math, 21:429--567, 1977.

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K. Appel and W. Haken. 1976. Every Planar Map is Four Colorable. Bulletin American Math Society,Vol 82, pp. 711-712.

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K. Appel and W. Haken. Every planar map is four colorable. Part ii. Reducibility. Illinois J. Math., 21:491--567, 1977.

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K. Appel and W. Haken. Every planar map is four colorable. Part i. Discharging. Illinois J. Math., 21:429--490, 1977.

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K. Appel, W. Haken, and J. Koch, Every planar map is four colorable. II. Reducibility. Illinois J. Math., 21(3):491-567, 1977.

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K. Appel and W. Haken, Every planar map is four colorable. I. Discharging. Illinois J. Math., 21(3):429-490, 1977.

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K. Appel, W. Haken, and J. Koch, Every planar map is four colorable. II. Reducibility. Illinois J. Math., 21(3):491-567, 1977.

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K. Appel and W. Haken, Every planar map is four colorable. I. Discharging. Illinois J. Math., 21(3):429-490, 1977.

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K. Appel, W. Hakenn, and J. Koch. Every planar map is four colorable. Part II. Reducibility. Illinois Journal of Mathematics, 21:491--597, 1977.

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K. Appel and W. Haken. Every planar map is four colorable. Part I. Discharging. Illinois Journal of Mathematics, 21:429--490, 1977.

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K. Appel, W. Haken, and J. Koch, Every planar map is four colorable. II. Reducibility. Illinois J. Math., 21(3):491-567, 1977.

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K. Appel and W. Haken, Every planar map is four colorable. I. Discharging. Illinois J. Math., 21(3):429-490, 1977.

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K. Appel and W. Haken, Every planar map is four colorable, AMS Contemporary Mathematics, 98, Amer. Math. Soc., Providence, RI, 1989.

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K. Appel, W. Haken and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J Mathematics, 21 (1977), 491--567.

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K. Appel and W. Haken, Every planar map is four colorable. Part I. Discharging, Illinois J Mathematics, 21 (1977), 429--490.

No context found.

Ken Appel and Wolfgang Haken, Every planar map is four-colorable, Bull. Amer. Math. Soc. 82 (1976), 711-712.

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K. Appel and W. Haken, Every Planar Map is Four Colorable, Amer. Math. Soc., Providence, R.I., 1989, 741 p.

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K. Appel, W. Haken, and J. Koch, Every planar map is four colorable. Part II: Reducibility, Illinois J. Math., 21 (1977), pp. 491--567.

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K. Appel and W. Haken, Every planar map is four colorable. Part I: Discharging, Illinois J. Math., 21 (1977), pp. 429--490.

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K. Appel and W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc., 82 (1976), pp. 711--712.

No context found.

Ken Appel and Wolfgang Haken, Every planar map is four-colorable, Bull. Amer. Math. Soc. 82 (1976), 711-712.

No context found.

K. Appel, W. Haken, and J. Koch, Every planar map is four colorable. II. Reducibility. Illinois J. Math., 21(3):491-567, 1977.

No context found.

K. Appel and W. Haken, Every planar map is four colorable. I. Discharging. Illinois J. Math., 21(3):429-490, 1977.

No context found.

K. Appel and W. Haken. Every planar map is four colorable. American Mathematical Society Bulletin, 82(5):711--712, 1976.

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K. Appel and W. Haken. Every planar map is four-colorable. Bull. Amer. math. Soc, 82:711-712, 1976.

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