exam-2-100407

# exam-2-100407 - C 3 and P 4 below C 3 P 4 P 4 ± C 3(a(5...

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MATH 682 Exam #2 Answer exactly four of the following six questions. Indicate which four you would like graded! 1. (10 points) Suppose that G is a simple graph that contains two edges whose removal destroys all cycles in G . Prove that G is planar. 2. (10 points) Let v 1 v 2 v 3 ∼ ··· ∼ v n be the longest path in a simple graph G . Show that χ ( G ) n . 3. (10 points) Prove that there is a tournament on n vertices with a directed Eulerian tour if and only if n is odd. 4. (10 points) The box product G ± H of two graphs G and H is a graph with vertices represented by ordered pairs ( u G ,u H ) where u G G and u H in H . ( u G ,u H ) is adjacent to ( v G ,v H ) if either u G = v G and u H v H or u G v G and u H = v H . As an example, we see the result of taking a box product of
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Unformatted text preview: C 3 and P 4 below: C 3 P 4 P 4 ± C 3 (a) (5 points) Prove that χ ( G ± H ) ≤ χ ( G ) χ ( H ). (b) (5 points) Prove that χ ( G ) + χ ( H )-2 ≤ χ ( G ± H ) ≤ χ ( G ) + χ ( H ) + 1. 5. (10 points) The cube Q 4 consists of sixteen vertices associated with the sixteen bit-strings 0000 , 0001 ,..., 1111. Two vertices are adjacent if they diﬀer in exactly one bit. Prove that Q 4 is nonplanar. 6. (10 points) Prove by construction that for n > 2 k , ex( C 2 k ,n ) ≥ 2 k ( n-2 k ). Page 1 of 1 April 8, 2010...
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