# hw04 - Math 412 HW4 Due Friday Students in the three credit...

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Math 412 HW4 Due Friday, February 26, 2016 Students in the three credit hour course must solve five of the six problems. Students in the four credit hour course must solve all six problems. 1 . Let T, T 0 be spanning trees of a connected graph G . For any e E ( T ) - E ( T 0 ) prove that there exists e 0 E ( T 0 ) - E ( T ) such that both T 0 + e - e 0 and T - e + e 0 are spanning trees of G . 2 . Let T be a tree on with an even number of vertices. Prove that T has exactly one spanning subgraph such that every vertex has odd degree. 3 . Given x V ( G ), let s ( x ) = v V ( G ) d ( x, v ). The barycenter of G is the set its vertices at which s ( x ) is minimized. a) Prove that the barycenter of a tree is a single vertex or two adjacent vertices. (Hint: Study s ( u ) - s ( v ) when uv E ( G ).) b) Give an example of a tree in which the distance between the center and the barycenter is at least 3. 4 . Using the Pr¨ufer correspondence, for n 6, count the number of trees with vertex set [ n ] that have maximum degree 3 and exactly four leaves. 5 . Let x be a vertex in a graph G , and suppose that
• Spring '13
• Dr.ZAre
• Graph Theory, Planar graph, credit hour course, connected graph, connected graph G.

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