Discrete Mathematics with Graph Theory (3rd Edition) 260

Discrete Mathematics with Graph Theory (3rd Edition) 260 -...

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258 (b) Yes. In the graph at the right, the bipartition sets are labeled {I, 2, 3, 4} and {A, B, C, D}. Solutions to Exercises 1 A :@: C 3 1 4 6. The labelings show that the graphs are isomorphic. 5 2 2 ~--\--h-+-_~B 1 E A C 7. [BB] Any graph 9 with n vertices is a subgraph of lCn , as is easily seen by joining any pair of vertices of 9 where there is not already an edge. 8. (a) [BB] Suppose that 9 and 1i are isomorphic graphs and r.p: V(Q) ---+ V(1i) is the isomorphism of vertex sets given by Definition 9.3.1. Label the edges of 9 arbitrarily, gl,g2, ... ,gm, and then use r.p to label the edges of as hI, h2, ... , hm; that is, if edge gl has end vertices v and w in 9, let hI be the edge joining r.p(v) and r.p(w) in 1i. Repeat to obtain h2"'" hm. Now a triangle in 9 is a set of three edges {gi, gj, 9k} each two of which are adjacent. It follows from the way we labeled the edges of that {gi, gj, gd is a triangle in 9 if and only if {hi, hj , hk} is a triangle in . Thus, the number of triangles in each graph is the same.
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