C OMBINATORIAL G RAPH T HEORY
Exercise Sheet 6
Prof. Dr. Ir. Gerhard Woeginger WS 2016/17
Tim Hartmann Due Date: December 5, 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 points)
The girthg(G)of a graphGis the length of its shortest cycle. A cycle-free graph has infinite girth.
• (a) Show that a planar graph with girthg <∞satisfies|E| ≤(n−2)·g/(g−2).
• (b) Deduce from (a) that the Petersen graph is non-planar.
Exercise 2 (4 points)
For a graphG = (V, E), its square G2 has the same vertex set V asG. Furthermore, the squareG2 has an edge between verticesuandv, if and only if inGthere exists a path of at most two edges betweenuandv.
Determine alln≥5for whichCn2
(that is, the square of the cycle onnvertices) is planar.
Exercise 3 (6 points)
A graph is outer-planar, if it allows a planar embedding in which all vertices lie on the boundary of the infinite face.
Prove that a graph is outer-planar, if and only if it does neither contain a subdividedK4nor a subdividedK2,3. Hint: Use Kuratowski’s theorem.
Exercise 4 (4 points)
Prove: Every3-connected non-planar graph onn≥6vertices contains a subdivision ofK3,3. Hint: Use Kuratowski’s theorem.
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