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3 . 4. ( ) v f e v + f = e + χ χ = v + f e D D A A a, b e, f, v χ χ 4 4 e, f, v a a a b b b 4 χ 5 n n n 5 5 χ = n + n = χ = v + f e Euler χ = χ = χ =
4 .. m n fm = vn = e = (m, n ) m n v = m n f, e = m f m n f + f = m f + f m n + > m (m + n) > mn m > (m )n m m > n, m m > 6 > m m m =, 4, 5 6 (m = ) m m = 4 (m = 4) (m = 5) m = n =, 4, 5 m = 4 n = m = 5 n = 5 4
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6 4 85 [Frederick Guthrie ] [A. de Morgan] [A.ayley] (A.Kempe) 89 (P. J. Heawood) n n 4. ( ) () () () (4), 4. ( ) 4. ( ) 4.4 Hint 4.5 n n [] [] [] n, n ayley. 5. 6
7 n f n f = l n= l v v = e, l nf n = e. e = v = n= l nf n. n= f n ( l ) ( l ) nf n nf n + n= n= l (6 n)f n =. n= l nf n =. (4.) 6 n > f n n= 7
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14 II f k [N] f f > [N] f = k [N] (5.) N(N ) 6(N χ) = ( N = 6 χ ) N χ = f + v e ( N = 6 f + v e ) N (5.) χ v e, i.e. v e (f + v e) N (f + v e) f = + e f v f + e f (5.) N e f (5.) N N Nf Nf e N e f Nf f, [N] A 4.7 A Q k [N] A [N] [N] A [N] ([N] = 7) = N 5.4 (Heawood) 7 Unger (5.) 6 [ ] 4
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17 ,, D + E A [] [] (978) [] [] K. Appel and W. Haken, Every Planar map is Four olorable, I. Discharging, Illinois J. of Math., (977) [] K. Appel and W. Haken, Every Planar map is Four olorable, II. Reducibility, Illinois J. of Math., (977) [4],, (985) [5] P. J. Heawood, Map-color theorems Quat. Journal of Math.,4(89)-8. [6],, lue ack, (978) [7] A.. Kempe, On the geographical problem of the four color, Amer J. of Math., (879)9-. [8],, (), (4) [9] Jun Mitani m/ 5, 7
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