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Discrete Mathematics with Graph Theory (3rd Edition) 126

# Discrete Mathematics with Graph Theory (3rd Edition) 126 -...

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124 Consider a (k + I)-gon. Let P be a vertex and Q, R the vertices adjacent to P. Join QR as shown on the right. Since k ?: 3, the line Q R separates the figure into two distinct pieces, triangle PQ R and the k-gon fonned by QR and all sides of the original (k + I)-gon except PQ and P R. By the induction hypothesis, the sum of the interior angles of this new k-gon equals (k - 2) 180 0 Also, triangle PQR has sum of interior angles equal to 180 0 Solutions to Exercises The sum of the angles of the (k + 1 )-gon is the sum of the angles in the k-gon and in the triangle, that is, (k - 2) 180 0 + 180 0 = (k - 1) 180 0 , as desired. By the Principle of Mathematical Induction, the result is true for all n ?: 3. 29. We prove this by induction on n, the number of straight lines drawn. If n = 1, we have two regions which, if colored with different colors, gives a proper coloring using just two colors. (By "proper," we mean that bordering countries have different colors.) Now suppose that k ?: 1 and the statement is true for n = k; that is, suppose that a map made by drawing k
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