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hw14sol

# hw14sol - Math 55 Discrete Mathematics Fall 2008 Homework...

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Math 55: Discrete Mathematics, Fall 2008 Homework 14 Solutions 8.6: 6. (a) and (c) are posets, (b) and (d) are not. 22 (c) 2 5 3 25 10 15 11 36 (a) ( R 0 , ) (b) ( R 0 , ) (c) ( R , ). 9.2: 18. Suppose there are n vertices. The possible degrees are then 0 through n - 1. If no two vertices have the same degree, there must be one of each degree. But if there is a vertex of degree n - 1, then every other vertex is adjacent to it, so there is no vertex of degree 0, if n 2. 26(a) K n is bipartite for n = 1 or 2. (b) C n is bipartite for n even. 34. It has (4 + 3 + 3 + 2 + 2) / 2 = 7 edges. * 58. Suppose the two parts have k and l vertices respectively. Then k + l = v , and e kl . Therefore v 2 / 4 - e ( k + l ) 2 / 4 - kl = ( k - l ) 2 / 4 0, so e v 2 / 4. 9.3: 38. Isomorphic, via u 1 7→ v 1 , u 2 7→ v 3 , u 3 7→ v 2 , u 4 7→ v 5 , u 5 7→ v 4 (other isomorphisms also exist). 44. Not isomorphic. One way to distinguish them is that in the v graph, the five vertices adjacent to each vertex v i contain a circuit of length 4, while in the u graph, they do not.

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