## The four-color problem: assaults and conquest |

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Page 77

This argument shows that u4 is D-reducible. Let us examine this Kempe-chain

aspect of D-reducibility a little more carefully. If we begin with a four-coloring c' of

Q, then each

This argument shows that u4 is D-reducible. Let us examine this Kempe-chain

aspect of D-reducibility a little more carefully. If we begin with a four-coloring c' of

Q, then each

**partition**of the four colors into two complementary pairs induces a ...Page 82

For any

the circuit Q is split into a/? and yd Kempe chains. Let q(n) denote the number of

these two-colored components. Lemma 3-5 If c' is any coloring of Q, c any ...

For any

**partition**n of the four colors into two complementary pairs, say n = aft/yd,the circuit Q is split into a/? and yd Kempe chains. Let q(n) denote the number of

these two-colored components. Lemma 3-5 If c' is any coloring of Q, c any ...

Page 87

For example, whenever q(nt) = 10 and q(7r2) = 8, the third

. But we really have enough information as it is. The graphs given in Fig. 3-27

suggest that Kempe-chain arguments have very small chance when x<0.10; very

...

For example, whenever q(nt) = 10 and q(7r2) = 8, the third

**partition**n has q(n) = 8. But we really have enough information as it is. The graphs given in Fig. 3-27

suggest that Kempe-chain arguments have very small chance when x<0.10; very

...

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### Contents

Historical Setting | 3 |

Map Coloring | 21 |

Solution of the FourColor Problem | 52 |

Copyright | |

7 other sections not shown

### Other editions - View all

The Four-color Problem: Assaults and Conquest Thomas L. Saaty,Paul C. Kainen No preview available - 1986 |

### Common terms and phrases

adjacent Appel and Haken argument assigned bipartite boundary bridgeless cubic c-colorable called chain group chromatic number chromatic polynomials chromials chromodendron cographic coloring of G complete graph components Conjecture C4 connected graph Corollary corresponding cubic graph cubic map D-reducible define denote directed graph disconnecting set disjoint dual dual graph edges of G embedding endpoints equivalent Euler's formula example Figure finite five-color following theorem four four-color conjecture four-color theorem four-coloring of G G contains G is planar graph G Graph Theory hamiltonian circuit hamiltonian graph Heawood Hence homeomorphic induced integer isomorphic Kempe chains Kempe residues Lemma Let G loop mathematical maximal planar graph minimum number neighbors number of colors number of edges number of vertices obtained pair partition plane problem Proof Let prove reducible configurations regions result subgraph of G subset surface sw(G Tait-coloring three-colored topological tree triangulation Tutte unavoidable set vertex of degree Whitney