Discrete Mathematics with Graph Theory (3rd Edition) 255

Discrete Mathematics with Graph Theory (3rd Edition) 255 -...

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Section 9.2 253 18. (a) [BB] No such graph exists. The sum of the degrees of the vertices is an odd number, 17, which is impossible. (b) No such graph exists: 100(101) 2 + 2 + 2 + 3 + 4 + . .. + 100 = 1 + (1 + 2 + 3 + . .. + 100) = 3 + 2 is odd. (c) [BB] Impossible. A vertex of degree 5 in a graph with six vertices must be adjacent to all other vertices. Two vertices of degree 5 means all other vertices have degree at least 2, but the given degree sequence contains a 1. (d) 0-0 0-0 0-0 (e) Not possible. Since there is a vertex of degree 5 there must be at least six vertices in the graph. (0 Not possible. The vertex of degree 5 would be joined to all other vertices. But now, the degrees of the two vertices of degree 1 have been accounted for, and it is not possible to have a vertex of degree 4. (g) Not possible. The vertices of degree 6 use up all the degrees 1, 1, 2, 2, 2, leaving a vertex of degree 4 impossible. (h) Yes. 19. No. Since there are five vertices, we would have to have at least three edges in order that every vertex
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Unformatted text preview: be incident with at least one of them, but then some vertex is incident with two edges, contradicting no adjacent edges. 20. (a) [BB] E degvi = 5(4) + 2(2) = 24 = 21£1, so there are 12 edges. (b) E deg Vi = 4(3) + 2(4) + 2(5) = 30 = 21£1, so there are 15 edges. 21. (a) [BB] This is not bipartite because it contains a triangle. (b) This is bipartite with bipartition sets indicated R and W on the graph at the left below. R W R W R BJ W R w~w R W R (c) [BB] This is bipartite with bipartition sets indicated R and W on the graph at the right above. (d) This is not bipartite. Consider the vertices labeled 1, 2, 3, 4, 5 in the picture at the left below. Vertices 1 and 2 are would have to lie in different bipartition sets and, thus, 1 and 3 would lie in the same set. Since 3 and 4, and 4 and 5 lie in different sets, 3 and 5 lie in the same set. But this...
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