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.
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