Discrete Mathematics with Graph Theory (3rd Edition) 260

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online