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CHAPTER TWO EULERs EQUATION
CHAPTER THREE HAMII TONIAN CIRCUITS
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3-coloring 3-connected 3-polyhedral 3-valent map 4-sided 5-valent vertex ac-chain argument boundary called Chapter coloring problems complete graph connected graph contain correspond coun countries meet duality edges and countries edges colored edges meet Euler characteristic Euler's equation example Exercise exist fewer edges five edges five or fewer four Four-Color Conjecture four-color problem Four-Color Theorem graph G graph isomorphic Hamiltonian circuit hedron inequality irreducible graph isomorphic maps Kempe chain Kempe's proof label least three edges map coloring map in Figure map is four-colorable Math mathematics minimum number multiple edges multiple of three number of colors number of countries number of edges number of marks number of vertices original graph original map pair planar graph polyhedron with exactly proper map prove reducible configurations regular polyhedra requiring five colors simplicial sphere square Suppose tetrahedron three colors tices torus triangles truncate unavoidable set vertices of valence