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Discrete Mathematics with Graph Theory (3rd Edition) 299

Discrete Mathematics with Graph Theory (3rd Edition) 299 -...

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Section 11.1 297 8. The six vertices of odd degree are {A, B, G, D, E, F} and the shortest path labels for all partitions are shown. {A, B}, {C, D}, {E, F} {A, B}, {C, F}, {D,E} {A, C}, {B, E}, {D, F} {A, D}, {B, C}, {E, F} {A, D}, {B,F}, {C,E} {A, E}, {B, D}, {C, F} {A, F}, {B, C}, {D,E} {A, F}, {B, E}, {C, D} 7+6+8 = 21 7+12+5 = 24 6+5+ 11 = 22 7+4+8 = 19 7+8+9=24 8+6+ 12 = 26 15+4+5 = 24 15+5+6 = 26 {AB,CE,DF} {AC,BD,EF} {AC,BF,DE} {AD,BE,CF} {AE,BC,DF} {AE,BF,CD} {AF,BD,CE} 7+9+11=27 6+6+8=20 6+8+5 = 19 7+5+12 = 24 8+4+11=23 8+8+6 = 22 15+6+9 = 30 So we see that one of two solutions is to add copies of edges AM, MG, BN, NO, OF, DP, PQ andQE. N 2 D 5 9. [BB] It certainly is possible and this is illustrated by the graph shown (all of whose vertices are odd). 10. [BB] Each even vertex in (} either has unchanged degree in (}' or, as an intermediate vertex on one or more paths between odd vertices, has its degree increased by a multiple of 2, so its degree remains even
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Unformatted text preview: in (}'. Each odd vertex in (} is the end vertex of precisely one new path constructed by the algorithm. In addition, an odd vertex could be an intermediate vertex on a path between odd vertices, so it is even in (}'. 11. Solution 1. The first odd vertex can be paired with any of the remaining n -1. For each pairing, one of the remaining vertices can be paired with any of n - 3 vertices. Then one of the remaining vertices can be paired with any of n - 5 and so on. The number of pairs is (n -l)(n -3)(n -5) ... 5·3· 1 Note that n is even so the product here really does finish as indicated. Solution 2. Let m = n/2. There are (~) ways to select a pair {Vb wd of odd vertices; for each of these ways, there are (n~2) ways to select another pair {V2' W2}, and so on. Altogether, there are ways to select {VI, WI}, ... , { Vm, wm}. Disregarding the order in which these pairs are selected, there are just 2::~! ways to divide the odd vertices into pairs....
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