The Four-color Problem: Assaults and Conquest |
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Page 42
... components X¡ , each of which is a maximal connected subset of S. If X - S is open , then since every point in X is euclidean , it can be shown that the connected components X ; of X are also open . Now suppose GS . By the above , the ...
... components X¡ , each of which is a maximal connected subset of S. If X - S is open , then since every point in X is euclidean , it can be shown that the connected components X ; of X are also open . Now suppose GS . By the above , the ...
Page 82
... components . Lemma 3-5 If c ' is any coloring of Q , c any feasible coloring of a comple- mentary configuration , and л a partition of the four colors , then ( a ) q ( л ) is even ; ( b ) X ( c , π ) has q ( л ) edges and q ( л ) + 1 ...
... components . Lemma 3-5 If c ' is any coloring of Q , c any feasible coloring of a comple- mentary configuration , and л a partition of the four colors , then ( a ) q ( л ) is even ; ( b ) X ( c , π ) has q ( л ) edges and q ( л ) + 1 ...
Page 146
... components of G. Lemma 6-15 There is a polynomial S ( G , 2 ) such that P ( G , λ ) = ( G ) S ( G , λ ) . PROOF Let k = k ( G ) and let G1 , ... , G be the components of G. Then k P ( G , λ ) = || P ( G1 , 2 ) i = 1 by Theorem 6-7 . By ...
... components of G. Lemma 6-15 There is a polynomial S ( G , 2 ) such that P ( G , λ ) = ( G ) S ( G , λ ) . PROOF Let k = k ( G ) and let G1 , ... , G be the components of G. Then k P ( G , λ ) = || P ( G1 , 2 ) i = 1 by Theorem 6-7 . By ...
Contents
Historical Setting | 3 |
Problems and Methods | 21 |
Solution of the FourColor Problem | 52 |
Copyright | |
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The Four-color Problem: Assaults and Conquest Thomas L. Saaty,Paul C. Kainen No preview available - 1986 |
Common terms and phrases
Academic Press adjacent bipartite boundary C₁ called chain group Chromatic Number chromatic polynomials chromials chromodendron cographic coloring of G Combinatorial complete graph components connected graph Corollary corresponding cubic graph cubic map D-reducible define denote disjoint dual dual graph edges of G embedded endpoints equivalent Euler's formula example Figure finite five-color following theorem Four Color Problem four-color conjecture four-color theorem four-coloring of G G contains G is planar G₁ G₂ graph G Graph Theory Graphen hamiltonian circuit Heawood Hence homeomorphic induced integer isomorphic k-colorable Kempe chains Kempe residues Lemma Let G Mathematics maximal planar graph minimum number neighbors number of colors number of edges number of vertices obtained pair partition plane PROOF Let prove reducible configurations regions result subgraph of G subset Suppose surface sw(G Tait-coloring three-colored topological tree triangulation Tutte unavoidable set v₁ v₂ vertex of degree Whitney