C OMBINATORIAL G RAPH T HEORY
Exercise Sheet 8
Prof. Dr. Ir. Gerhard Woeginger WS 2016/17
Tim Hartmann Due Date: January 9, 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 (2+2 points)
Determine the bandwidth of
• (a) the complete bipartite graphK2a,2bwitha, b≥1;
• (b) the complete tri-partite graphK2c,2c,2cwithc≥1.
Exercise 2 (3 points)
For any graphG= (V, E), prove that bw(G+e)≤2bw(G).
Exercise 3 (3 points)
Consider a drawing of a planar graphG= (V, E), in which all faces are triangles, except for the four-sided infinite faceabcd. LetV =V1∪V2be a partition ofV witha, c∈V1andb, d∈V2. Prove that either inV1there is a path from atoc, or inV2there is a path frombtod
Hint: You may use Sperner’s lemma.
Exercise 4 (6 points)
• (a) In a drawing of a planar graphG, all faces have an even number of edges. Prove thatGis bipartite.
• (b) In a drawing of a planar graphG, all vertices have even degree. Prove that the faces can be2-colored so that faces that share a common edge are always colored differently.
• (c) Decide whether there exists a drawing of some planar graph with the following properties: The infinite face is a pentagon, and all other faces are triangles, and all vertices have even degree.
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