FCS 2016 Midterm 2.
November 22th, 2016
Each problem is worth 10 points. One needs 18 points to pass the test. Results without proper reasoning receive no credit. You can only use pen or pencil and paper. Usage of calculators, mobile phones or other electronic devices, furthermore notes is forbidden. It is also strictly forbidden to seek help from others or to give help to others. Violation of the rules results in disqualification from the midterm.
Write your name and Neptun code IN PRINT on each page you turn in!
Problems
1. Let G be a graph with 100 vertices and assume that its chromatic number χ(G) = 15.
Furthermore, let us assume that G has 8 vertices that are pairwise adjacent to each other and also these 8 vertices are adjacent to all other vertices of the graph. Prove that the size of the largest independent vertex set satisfies α(G)≥14.
2. Show a minimum capacity cut in the network (G, s, t, c) shown on the picture on the left.
3. Let us assume that bipartite graph G = (V, E) has 100 vertices and the size of largest independent vertex set is α(G) = 50. Show that for any subset X ⊆ V of the vertices of G, |N(X)| ≥ |X| holds.
4. Is the graph on the picture on the right planar?
s
a b c
t
f 22
11
22
33 7
41 9
9 15
11
25
33 20
10
e d
5. Suppose that for a finite simple graph G, τ(G) = ω(G)−1 holds for the sizes of minimal vertex cover and maximal clique. Prove that the clique number and the chromatic number of G are equal, that is χ(G) =ω(G).
6. Today is Patricia’s birthday. During math class, the teacher told her that three years ago her age was a divisor of the actual number of the year. How old is Patricia?
