Full answer: Assign a grade of A (correct), C (partially correct), or F (failure) to

Chapter 3, Problem 15

(choose chapter or problem)

Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify assignments of grades other than A.(a) Claim. If v and w are vertices in a graph such that andthenProof. Suppose and Then there is a pathfrom u to v with length 3 and a path from uto w with length 4. Then is a path from v to wwith length 7. The distance from v to w is the length of the shortest pathfrom v to w and there is a path of length 7, so(b) Claim. Every connected graph G of order n has a closed path of order n.Proof. Let G be connected graph of order n with verticesSince G is connected, each of the vertices is reachable fromand is reachable from Thus connecting these paths there is apath from to to to and back to By Theorem 3.5.2(b)the length of any closed path, including this one, is at most n.