By L. Bostock, F.S. Chandler

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Our proof is based on a process of removing an edge or an edge and a vertex from a connected graph so that the resulting graph is still connected. We will then show that V - E + F does not change during this process, and finally we show that through repeated use of this process we arrive at a graph for which V - E + F == 2. This will clearly show that Euler's equation holds for the original graph. The process consists of doing one of two things: either removing an edge, but leaving both of its endpoints, or removing an edge and one of its endpoints.

When we remove a I-valent vertex and its edge, we decrease V and E 28 MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM by one while leaving F unchanged, thus V - E + F is unchanged. When we remove an edge and leave its two endpoints, we decrease E by one. Since the removed edge is on two different faces (one inside the circuit and one outside the circuit), and since these two faces become one face after the removal, F is decreased by one. Again, V - E + F is left unchanged. Now observe what happens when we do this process repeatedly to a given planar graph.

Your results should correspond to the last four lines of Table 1. v E F TETRAHEDRON 4 6 4 CUBE 8 12 6 OCTAHEDRON 6 12 8 DODECAHEDRON 20 30 12 ICOSAH E DRO N 12 30 20 TABLE 1 You should not get the impression that we have proved the existence of any regular polyhedra. What we have done is proved that if a regular polyhedron of a given type exists, then it must have a certain number of vertices, edges and faces. We have alnzost given a proof that there can be at most five regular polyhedra. Many authors present an argument like ours as if it were a proof, but it is not.