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Unformatted text preview: now we must trace over some edges more than
once.
(a) (10 points) Prove that regardless of the graph, it is impossible to trace the graph without going
over at least m edges more than once. (Note that there are graphs for which the number of edges
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that must be traced more than once could be very large, and much larger than m.)
Solution: During a tracing, each vertex must be entered as many times as it is exited. The only
way to do this at an odd degree vertex is to retrace at least one edge. Each retraced edge is
incident to at most two odd degree vertices, so there must be at least m retraced edges in order
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to have one incident to each of m vertices. (b) (5 points) Give an example of a graph with m = 4 odd degree vertices which you can trace with
exactly m edges traced over more than once (twice). Clearly mark the m = 4 odd degree vertices
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and the m = 2 edges you trace twice.
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Solution: The odd degree vertices are circled, and the edges used twice are bolded.
a d CS 70, Fall 2013, Midterm #2 b c 6...
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This document was uploaded on 01/28/2014.
 Winter '09

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