C OMBINATORIAL G RAPH T HEORY
Exercise Sheet 3
Prof. Dr. Ir. Gerhard Woeginger WS 2016/17
Tim Hartmann Due Date: November 14, 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)
LetGbe a graph with verticesv1, . . . , vn, and letdibe the degree of vertexvi. Show thatGcontains an independent set of at leastPn
i=1 1
di+1 vertices.
Exercise 2 (2+1 points)
(a) LetG= (V, E)be a graph withnvertices,medges, andttriangles. Prove thatt≥m(4m−n2)/(3n).
Hint: Denote the number of triangles that contain the edge[u, v]bytuv. Show thatd(u) +d(v)≤n+tuv, and investigate the sum of all these inequalities.
(b) Deduce that a graph with m ≥ (14 +ε)n2 edges containsΩ(εn3)triangles. In other words: If the threshold in Mantel’s theorem is increased by just a little bit, then the resulting graph contains a constant fraction of all possible triangles.
Exercise 3 (5+1 points)
(a) LetGbe a graph withn≥4vertices andm > n2/4edges. Show thatGhas a vertexv, whose removal leaves a graphG−vwith more than(n−1)2/4edges.
(b) Use the statement in (a) to derive an inductive proof of Mantel’s theorem.
Exercise 4 (4 points)
For two pointsP1 = (x1, y1)andP2 = (x2, y2)in the Cartesian plane, we denote their Manhattan distance (or: `1- distance) byd(P1, P2) = |x1−x2|+|y1−y2|. If1 ≤ d(P1, P2) < 2holds, then these two points form a so-called RWTH-pair.
Determine the maximum number of (unordered) RWTH-pairs that can occur in a set of 24 points.
Hint: Use Tur´an’s theorem.
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