Combinatorial Graph Theory
Exercise Sheet 10
Prof. Dr. Ir. Gerhard Woeginger WS 2016/17
Tim Hartmann Due Date: January 23, 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 earn 50% of the sum of all points.
• You can earn 50% 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 (3 points)
In a split graphG= (V, E), the vertex set V is partitioned intoC and I. The edge setE contains all edges between vertices inC(so thatC induces a clique), none of the edges between vertices inI (so thatI induces an independent set), and some of the edges betweenC andI.
Prove that every split graph Gis χ(G)-choosable.
Exercise 2 (2+2 points)
Decide whether the following graphs are 2-choosable:
(a) A tree on 100 vertices.
(b) The 6-vertex graph that results from glueing together two cycles C4 along an edge.
Exercise 3 (4 points)
Prove that the complete bipartite graphKs,t withs≤t iss-choosable if and only if t < ss.
Exercise 4 (5 points)
Forn≥3, the graphGn results by identifying the end-vertices of the three pathsP3,P3, and Pn (wherePk
is the path onk vertices andk−1 edges).
Determine all integersn≥3 for whichGn is 2-choosable.
1
