## The Complexity of the Four Colour Theorem (2009)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Calude09thecomplexity,

author = {Cristian S. Calude and Elena Calude},

title = {The Complexity of the Four Colour Theorem},

year = {2009}

}

### OpenURL

### Abstract

The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq. In this paper we use the computational method for evaluating (in a uniform way) the complexity of mathematical problems presented in [8, 6] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem has roughly the same complexity as the Riemann hypothesis and almost four times the complexity of Fermat’s last theorem. 1

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Citation Context ...e We briefly describe the syntax and the semantics of the register machine language which implements a (natural) universal prefix-free Turing machine (it is a refinement of the languages described in =-=[11, 8]-=-) which is used to obtain the program Π4CT. Any register machine has a finite number of registers, each of which may contain an arbitrarily large non-negative binary integer. The list of instructions ... |

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Citation Context ...f every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. Shortly, every planar graph is four-colourable. The theorem was proved in 1977 =-=[1, 2]-=- (see also [17]) using a computer-assisted proof which consists in constructing a finite set of “configurations”, and verifying that each of them is “reducible”—which implies that no configuration wit... |

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Citation Context ... of mapmaking do not mention the four-color property.” 2 A diophantine equational presentation of the four colour property We use the Diophantine representation of the four colour theorem proposed in =-=[12]-=-, i.e. a Diophantine equation F (n, t, a, . . .) =0, (1) 1 This computer-assisted proof generated much mathematical and philosophical discussions around the notion of acceptable mathematical proof, se... |

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Citation Context ... measure of complexity of the four colour theorem. How? Simply by counting the number of bits necessary to specify Π4CT in some fixed “universal formalism” (a universal self-delimiting Turing machine =-=[5]-=-). Of course, there 5 The integer remainder function is denoted by rem. 3are many programs equivalent to Π4CT, so a natural way to evaluate the complexity is to consider the smallest such program [8]... |

21 |
Every planar map is four colourable
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(Show Context)
Citation Context ...f every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. Shortly, every planar graph is four-colourable. The theorem was proved in 1977 =-=[1, 2]-=- (see also [17]) using a computer-assisted proof which consists in constructing a finite set of “configurations”, and verifying that each of them is “reducible”—which implies that no configuration wit... |

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Citation Context ...rated much mathematical and philosophical discussions around the notion of acceptable mathematical proof, see for example [4, 9, 10]. 2 It appears that there is no verification in its entirety. 3 See =-=[13]-=- for a recent presentation of the formal proof. 4 Our emphasis. 2such that (1) has no solution if and only if every planar graph can be coloured with at most four colours so that no two adjacent vert... |

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Citation Context ...of the theorem was obtained with the equational logic program Coq. In this paper we use the computational method for evaluating (in a uniform way) the complexity of mathematical problems presented in =-=[8, 6]-=- to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem has roughly the same complexity as ... |

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Citation Context ...antine equation F (n, t, a, . . .) =0, (1) 1 This computer-assisted proof generated much mathematical and philosophical discussions around the notion of acceptable mathematical proof, see for example =-=[4, 9, 10]-=-. 2 It appears that there is no verification in its entirety. 3 See [13] for a recent presentation of the formal proof. 4 Our emphasis. 2such that (1) has no solution if and only if every planar grap... |

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Citation Context ...aph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. Shortly, every planar graph is four-colourable. The theorem was proved in 1977 [1, 2] (see also =-=[17]-=-) using a computer-assisted proof which consists in constructing a finite set of “configurations”, and verifying that each of them is “reducible”—which implies that no configuration with this property... |

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Citation Context ...antine equation F (n, t, a, . . .) =0, (1) 1 This computer-assisted proof generated much mathematical and philosophical discussions around the notion of acceptable mathematical proof, see for example =-=[4, 9, 10]-=-. 2 It appears that there is no verification in its entirety. 3 See [13] for a recent presentation of the formal proof. 4 Our emphasis. 2such that (1) has no solution if and only if every planar grap... |

3 |
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(Show Context)
Citation Context ...of the theorem was obtained with the equational logic program Coq. In this paper we use the computational method for evaluating (in a uniform way) the complexity of mathematical problems presented in =-=[8, 6]-=- to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem has roughly the same complexity as ... |