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hw11soltext

# hw11soltext - Math 55 Discrete Mathematics UC Berkeley...

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UC Berkeley, Spring 2009 Solutions to Homework # 11 (due April 27) § 9.1, #13: See diagrams. § 9.1, #27: This is best represented by a directed multigraph which allows loops but not duplicate edges. § 9.2, #5: No, because X v V deg v = 45 which is odd; but this is impossible by the Handshaking Theorem. § 9.2, #20: See diagrams. § 9.2, #29: a) K n has n vertices and ( n 2 ) edges. b) C n has n vertices and n edges. c) W n has N + 1 vertices and 2 n edges. d) K m,n has m + n vertices and mn edges. e) Q n has 2 n vertices and n 2 n - 1 edges (since each vertex has degree n ). § 9.2, #31: a) 3,3,3,3 b) 2,2,2,2 c) 4,3,3,3,3 d) 3,3,2,2,2 e) 3,3,3,3,3,3,3,3 § 9.2, #36: a) Not graphic; the vertex with degree 5 must be adjacent to each of the others, but one of these has degree 0. b) Not graphic; a simple graph with 6 vertices cannot have a vertex of degree 6. c) The cycle C 6 is one such graph. d) Not graphic: the degrees sum to an odd number, violating the Hand- shaking Theorem. e) See diagrams. f) See diagrams. g) The wheel W 5 is one such example. 1

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hw11soltext - Math 55 Discrete Mathematics UC Berkeley...

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