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Solution of the FourColor Problem
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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