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Unformatted text preview: MATH 682 Exam #1 Answer exactly four of the following six questions. Indicate which four you would like graded! 1. (10 points) Answer the following questions related to bound-subverting examples: (a) (4 points) It is known that the connectivity ( G ) of a graph G is bounded above by the minimum degree ( G ). Describe a method of constructing a connected graph with arbitrarily large but with a fixed, small . (b) (6 points) Let a flow f on a weighted digraph D be called addition-maximal if there is no valid flow g 6 = f such that g ( e ) f ( e ) for all edges; in other words, there is no valid flow resulting solely from adding flow to f . Describe (with an example, if such is useful) the construction of a weighted digraph and addition- maximal flow f such that | f | = 1 but D has maximal flow k for an arbitrarily large integer k . 2. (10 points) Let G be a graph containing a cycle C , such that there is a path P of length k between two vertices of G . Show that G contains a cycle of length...
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This note was uploaded on 01/12/2012 for the course MATH 682 taught by Professor Wildstrom during the Spring '09 term at University of Louisville.
- Spring '09