Discrete Mathematics with Graph Theory (3rd Edition) 300

Discrete Mathematics with Graph Theory (3rd Edition) 300 -...

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298 Solutions to Exercises 12. (a) Let VI be any odd vertex. Since VI acquires even degree in gil, it is incident with a new edge VI W. If W is odd, then VI W is the desired path. Otherwise, W is even in g and, still even in gil, and hence incident with another new edge wu. If u is odd, then VI wu is the desired path. Otherwise, continue until the first odd vertex WI is reached. (b) Continuing as in (a), let PI be a path of new edges from VI to WI and suppose there is another odd vertex v2 (rt {VI, WI}). Apply the procedure outlined in (a) to this vertex, but with two changes. If the path of new edges starting at V2 reaches a vertex W on PI which was even in g, then there must be another new edge incident with W (since W still has even degree in gil). Follow from W that edge which was not in Pl. Also, if the first odd vertex encountered on the path of new edges from V2 should happen to be VI (or WI or V2), then observe that there must be a third new edge incident with VI (WI, V2 respectively) because VI has even degree in
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