## An Update on the Four-Color Theorem (1998)

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Citations: | 23 - 5 self |

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@MISC{Thomas98anupdate,

author = {Robin Thomas},

title = {An Update on the Four-Color Theorem},

year = {1998}

}

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### Abstract

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1289 |
Graph Theory with Applications
- Bondy, Murty, et al.
- 1976
(Show Context)
Citation Context ...e e such that the graph Gne obtained from G by deleting e has more connected components than G. It is easy to see that if a cubic graph has an edge 3-coloring, then it has no cut-edge. Tait (see also =-=[3]-=- or [4]) showed in 1880 that the 4CT is equivalent to the following. THEOREM 3. Every cubic plane graph with no cut-edge has an edge 3-coloring. The equivalence of Theorems 1 and 3 is not hard to see,... |

729 |
Graph Theory
- Diestel
- 2000
(Show Context)
Citation Context ...h that the graph Gne obtained from G by deleting e has more connected components than G. It is easy to see that if a cubic graph has an edge 3-coloring, then it has no cut-edge. Tait (see also [3] or =-=[4]-=-) showed in 1880 that the 4CT is equivalent to the following. THEOREM 3. Every cubic plane graph with no cut-edge has an edge 3-coloring. The equivalence of Theorems 1 and 3 is not hard to see, and ca... |

283 |
Every Planar Map is Four Colorable
- Appel, Haken
- 1989
(Show Context)
Citation Context ...ppel and Haken (abbreviated A&H), when they published their proof of the Four Color Theorem in two 1977 papers, the second one joint with Koch. An expanded version of the proof was later reprinted in =-=[1]-=-. Let me state the result precisely. Rather than trying to define maps, countries, and their boundaries, it is easier to restate Guthrie's 1852 conjecture using planar duality. For each country we sel... |

116 | The four-colour theorem
- Robertson, Sanders, et al.
- 1997
(Show Context)
Citation Context ...e I turn to a more detailed discussion of configurations, reducibility and discharging, let me say a few words about the use of computers in our proof. The theoretical part is completely described in =-=[7]-=-, but it relies on two results that are stated as having been proven by a computer. The rest of [7] consists of traditional (computer-free) mathematical arguments. There is nothing extraordinary about... |

39 |
Hadwiger’s conjecture for K6-free graphs, Combinatorica 13
- ROBERTSON, SEYMOUR, et al.
- 1993
(Show Context)
Citation Context ...ult preceded the proof of the 4CT by four decades and, in fact, inspired Hadwiger’s conjecture.) Recently, Robertson, Seymour, and I were able to show that the next case is also equivalent to the 4CT =-=[8]-=-. More precisely, we managed to prove (without using the 4CT) the following. Theorem 17. Let G be a counterexample to Hadwiger’s conjecture for t =5 with the minimum number of vertices. Then G is apex... |

19 |
Map coloring and the vector cross product
- KAUFFMAN
- 1990
(Show Context)
Citation Context ...ch that the evaluations of the two associations are equal. This is easy to do by choosing v 1 = v 2 = \Delta \Delta \Delta = v k . But how about making the two evaluations equal and nonzero? Kauffman =-=[5]-=- has shown the following. THEOREM 4. Let i; j; k be the usual unit vector basis of R 3 . If two associations of v 1 \Theta v 2 \Theta : : : \Theta v k are given, there exists an assignment of i; j; k ... |

13 |
Thirteen colorful variations on Guthrie’s four-color conjecture
- SAATY
- 1972
(Show Context)
Citation Context ...s, we have found that just the opposite is the case. The four color problem and the generalization discussed here is central to the intersection of algebra, topology and statistical mechanics." S=-=aaty [10]-=- presents 29 equivalent formulations of the 4CT. In this article, let me repeat the most classical reformulation, and then mention three new ones. A graph is cubic if every vertex has degree three; th... |

5 |
Lie algebras and the four color theorem, Combinatorica 17
- BAR-NATAN
- 1997
(Show Context)
Citation Context ...her, it can be shown that WL (G) is a polynomial in N of degree at most k = 1 2 jV (G)j + 2, and so we can define W top L (G) to be the coefficient of N k in WL (G). The next result, due to Bar-Natan =-=[2], is best -=-introduced by a quote from his paper: "For us who grew up thinking that all there is to learn about sl(N) is already in sl(2), this is not a big surprise:" THEOREM 6. For a connected cubic g... |

5 |
Tutte’s edge-coloring conjecture
- Robertson, Seymour, et al.
- 1997
(Show Context)
Citation Context ...of G, and we say that a graph is doublecross if it can be drawn in the plane with two crossings in such a way that the two crossings belong to the same region (see Figure 9). Robertson, Seymour and I =-=[9]-=- have shown THEOREM 15. Let G be a counterexample to conjecture 14 with jV (G)j minimum. Then G is apex or doublecross. Figure 9. An apex and a doublecross graph. Thus, in order to prove Conjecture 14... |

4 |
Hadwiger's conjecture for K 6 -free graphs, Combinatorica 13
- Robertson, Seymour, et al.
- 1993
(Show Context)
Citation Context ...ult preceded the proof of the 4CT by four decades, and, in fact, inspired Hadwiger's conjecture.) Recently, Robertson, Seymour and I were able to show that the next case is also equivalent to the 4CT =-=[8]-=-. More precisely, we managed to prove (without using the 4CT) THEOREM 17. Let G be a counterexample to Hadwiger's conjecture for t = 5 with the minimum number of vertices. Then G is apex. By the 4CT e... |

3 | The Four Colour Theorem as a possible corollary of binomial summation, manuscript
- MATIYASEVICH
(Show Context)
Citation Context ...re otherwise. The two results combined with Theorem 3 immediately establish Theorems 6 and 7. The details may be found in [2]. The last reformulation, in terms of divisibility, is due to Matiyasevich =-=[6]-=-. THEOREM 8. There exist linear functions A k , B k , C k and D k (k = 1; 2; : : : ; 986) of 21 variables such that the Four Color Theorem is equivalent to the statement that for every two positive in... |

1 |
edge-coloring conjecture
- Tutte’s
- 1997
(Show Context)
Citation Context ... G , and we say that a graph is doublecross if it can be drawn in the plane with two crossings in such a way that the two crossings belong to the same region (see Figure 9). Robertson, Seymour, and I =-=[9]-=- have shown the following. Theorem 15. Let G be a counterexample to conjecture 14 with jV (G)j minimum. Then G is apex or doublecross. Thus, in order to prove Conjecture 14 it suffices to prove it for... |