Discrete Mathematics with Graph Theory (3rd Edition) 354

Discrete Mathematics with Graph Theory (3rd Edition) 354 -...

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352 Solutions to Exercises k + 1 ~ r + 1. Thus the number of edges on the path UVl V2 ... VrV is r 2': k, which is precisely the fact we wanted to show. 14. Let the bipartition sets of JC2 ,n be {Ul' U2} and {v!, V2, . .. ,v n }. To show that all spanning trees are obtained as depth-first search spanning trees with n = 2,3,4 we could simply list the possible trees. For example, the four spanning trees of JC2 ,2 are shown to the right, and these are all depth-first search trees. This approach is laborious, however, especially when n = 4, so we will give a general argument instead. Any spanning tree of JC 2,n must have some Vi adjacent to both Ul and U2, and the remaining Vj are partitioned into those adjacent to Ul and those adjacent to U2. If n = 3 and the two other vertices (different from Vi) are both adjacent to the same Ui, say U2, we get a tree of the type shown. This tree comes from a depth-first search starting at Ul. If
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