## Graph theory: a development from the 4-color problem |

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1-factor 2-block 2-connected 4-color conjecture 4-color problem adjacent arbitrary graph bipartite graph Birkhoff boundary bridges called chain group Chapter chromatic number chromatic polynomial circuit of length clearly cocycle cographic colors complete bipartite graph complete graph components connected graph corresponding country F cut vertex D-reducible decomposes degree denote disjoint dual graph edge set embedded endpoints Euler exactly example exists Figure G contains G is connected graph G graph theory Hamiltonian circuit Heawood Hence holds independent sets induction interior irreducible isomorphic joined Jordan curve Kempe least LEMMA Let G lf G loops matrix matroid Menger's Theorem minimal number multiple edges nonorientable normal cubic map number of edges number of vertices obtain path Petersen graph planar plane graph polygon polytopal precisely proof prove reducible configurations resp result separating vertex set Show simple graph subspace surface Tait's conjecture THEOREM triangulation Tutte unavoidable set vectors verify W-dual Whitney