Discrete Mathematics with Graph Theory (3rd Edition) 264

Discrete Mathematics with Graph Theory (3rd Edition) 264 -...

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262 Solutions to Review Exercises Proof. ( f- ) If n = 2m, the complete bipartite graph Kd,d has the required property. ( f- ) Now assume that 9 is a bipartite graph with n vertices each of degree d. Label the bipartition sets VI and V2, and suppose they contain nl and n2 vertices, respectively. Now count the edges in 9 and note that each edge contributes 1 to the sum of the degrees of the vertices in VI and one to the sum of the degrees of the vertices in V2. Thus, we discover dnl = L degv = L degv = dn2, VEVl VEV2 so nl = n2 and n = nl + n2 is even. Now a vertex v in VI is adjacent only to vertices in V2, so deg v = d ::; n2 = ~. This completes the argument. 10. (a) Assume this is possible. Then two of the vertices of would have to belong to the same bipartition set in 9 (since has at least three vertices). In a bipartite graph, however, vertices in the same bipartition set are not adjacent. This contradicts the fact that is complete, so no such situation is possible. (b) This is indeed possible. For example, let
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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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