The Method of Mathematical Induction by I. S. Sominskii

Author:I. S. Sominskii
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2019-03-08T16:00:00+00:00

Fig. 31

Case 4. σ has five boundaries. In this case, except for permutations of the digits 1, 2, 3, there is only one possible way of numbering the boundaries around the country σ (Fig. 32a). Let us examine first the case in which country σ5 neither coincides with σ2 or σ3 nor has boundaries in common with them. Let us join country σ5 to a, assign the number 2 to the new boundary MB, the number 1 to the new boundary RD, and change the number of the boundary BC to 1, and of the boundary CD to 2. In this way we obtain the normal map S' (Fig. 32b) with properly numbered boundaries. Since the number of countries in S' is n, it is possible to color it properly with four colors. If the country σ' = σ + σ5 is colored a, then the countries σ2' and σ4' will be colored β, and the countries σ1' and σ3' will be colored γ. Restoring the country a, we color it 8.

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