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Quantum Interpretations of the Four Color Theorem - Kainen (2001) (Correct)
....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.
Quantum Interpretations of the Four Color Theorem - Kainen (2001) (Correct)
....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.
A Compiled Implementation of Strong Reduction - Grégoire, Leroy (2002) (3 citations) (Correct)
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K. Appel and W. Haken. Every planar map is four colorable. Illinois J. Math, 21:429--567, 1977.
Hunter-Gatherer: Applying Constraint Satisfaction.. - Beale (1997) (6 citations) (Correct)
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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.
Generic Decentralized Control for Lattice-Based.. - Butler, Kotay, Rus.. (Correct)
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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.
Generic Decentralized Control for Lattice-Based.. - Butler, Kotay, Rus.. (Correct)
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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.
Three-Dimensional Orthogonal Graph Drawing - Wood (2000) (3 citations) (Correct)
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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.
Three-Dimensional Orthogonal Graph Drawing - Wood (2000) (3 citations) (Correct)
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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.
Three-Dimensional Orthogonal Graph Drawing - Wood (2000) (3 citations) (Correct)
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.
Three-Dimensional Orthogonal Graph Drawing - Wood (2000) (3 citations) (Correct)
No context found.
K. Appel and W. Haken, Every planar map is four colorable. I. Discharging. Illinois J. Math., 21(3):429-490, 1977.
Computer Assisted Proof of Optimal Approximability Results - Zwick (2002) (1 citation) (Correct)
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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.
Computer Assisted Proof of Optimal Approximability Results - Zwick (2002) (1 citation) (Correct)
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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.
Three-Dimensional Orthogonal Graph Drawing - Wood (2000) (3 citations) (Correct)
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.
Three-Dimensional Orthogonal Graph Drawing - Wood (2000) (3 citations) (Correct)
No context found.
K. Appel and W. Haken, Every planar map is four colorable. I. Discharging. Illinois J. Math., 21(3):429-490, 1977.
Some Probabilistic Restatements of the Four Color Conjecture - Matiyasevich (2003) (Correct)
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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.
Some Probabilistic Restatements of the Four Color Conjecture - Matiyasevich (2003) (Correct)
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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.
Some Probabilistic Restatements of the Four Color Conjecture - Matiyasevich (2003) (Correct)
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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.
Graph Color Extensions: - When Hadwiger's Conjecture (Correct)
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Ken Appel and Wolfgang Haken, Every planar map is four-colorable, Bull. Amer. Math. Soc. 82 (1976), 711-712.
One Probabilistic Equivalent of the Four Color Conjecture - Matiyasevich (2003) (Correct)
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K. Appel and W. Haken, Every Planar Map is Four Colorable, Amer. Math. Soc., Providence, R.I., 1989, 741 p.
One Probabilistic Equivalent of the Four Color Conjecture - Matiyasevich (2003) (Correct)
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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.
One Probabilistic Equivalent of the Four Color Conjecture - Matiyasevich (2003) (Correct)
No context found.
K. Appel and W. Haken, Every planar map is four colorable. Part I: Discharging, Illinois J. Math., 21 (1977), pp. 429--490.
One Probabilistic Equivalent of the Four Color Conjecture - Matiyasevich (2003) (Correct)
No context found.
K. Appel and W. Haken, Every planar map is four colorable, Bull. Amer. Math. Soc., 82 (1976), pp. 711--712.
Graph Color Extensions: When Hadwiger's Conjecture and.. - Albertson, Hutchinson (2002) (Correct)
No context found.
Ken Appel and Wolfgang Haken, Every planar map is four-colorable, Bull. Amer. Math. Soc. 82 (1976), 711-712.
Three-Dimensional Orthogonal Graph Drawing - Wood (2000) (3 citations) (Correct)
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.
Three-Dimensional Orthogonal Graph Drawing - Wood (2000) (3 citations) (Correct)
No context found.
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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