Discrete Mathematics with Graph Theory (3rd Edition) 370

# Discrete - 368 1 Solutions to Exercises 2~2\il5JI 1 3(b[BB The converse o f the Four-Color Theorem states that i f the chromatic number o f a graph

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368 Solutions to Exercises 1 2~2 \il5JI 1 3 (b) [BB] The converse of the Four-Color Theorem states that if the chromatic number of a graph is at most four, then the graph is planar. This result is false. The Petersen graph is not planar (Exercise 4 of Section 13.1), but, as we saw in part (a), X = 3. 7. (a) [BB] Yes, a tree is planar and we can prove this by induction on n, the number of vertices. Cer- tainly a tree with one vertex is planar. Then, given a tree with n vertices, removing a vertex of degree 1 (and the edge with which it is incident) leaves a tree with n -1 vertices which is planar by the induction hypothesis. The deleted vertex and edge can now be reinserted without destroying planarity. (b) Since a tree is connected and planar, we must have V - E + R = 2. But R = 1, hence, V - E = 1 or E = V-I, as desired. (c) The proofthat X(T) = 2 is perhaps easiest by induction on n, the number of vertices. If n = 2, the tree must be 0-0 and this can be colored with 2 colors, but not with fewer. Assuming the
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## This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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