C OMBINATORIAL G RAPH T HEORY
Exercise Sheet 2
Prof. Dr. Ir. Gerhard Woeginger WS 2016/17
Tim Hartmann Due Date: November 7, 16:15
Department of Computer Science RWTH Aachen University
• Hand in your solutions in a group of two or three students.
• Write the name and enrollment number of each group member on every sheet that you hand in.
• To achieve the permission for the exam you must earn50%of the sum of all points.
• You can earn50%bonus points by presenting your solution. At the beginning of every exercise session, you can mark the exercises that you want to present.
• If a student is not able to present a correct solution although he/she marked the exercise as presentable, he/she will lose all of his/her points on the exercise sheet.
Exercise 1 (1+1 points)
Characterize all trees over the vertex set{1,2, . . . , n}
(a) whose Pr ¨ufer Code consists of(n−2)-times the same number (b) whose Pr ¨ufer Code consists ofn−2pairwise distinct numbers
Exercise 2 (4 points)
The graphKn−e(speak:Knminus an edge) results from the complete graphKnonn≥3vertices by removing one edge. Determine the number of spanning trees inKn−e.
Exercise 3 (4 points)
Prove: For every integerq≥3, there exists a graphGq on2q−1vertices that has exactlyq2spanning trees.
Exercise 4 (4 points)
Use the matrix-tree-theorem to determine the number of spanning trees in the complete bipartite graphKr,s with r, s≥1.
Hint: Remember the characteristic polynomial from your linear algebra class.
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