FCS 2017 Midterm 2.
November 30th, 2017
Each problem is worth 10 points. One needs 18 points to pass the test, 48 points on the two midterms together to obtain signature. 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. Determine the chromatic number of the graph shown on the figure below.
2. What is the maximum flow in the network below if we assume that parameter p= 0? How does the maximum flow value depend on p if p > 0?
b
17 33 22
11
s 11 t
d
c p
15 37
9 a
25
3. Let the color classes of bipartite graph G be A = {a, b, c, d, e, f} and B = {1,2,3,4,5,6}, respectively, and let the edges be a1, a2, a3, b2, b4, c2, c3, c5, d2, d4, e2, e4, f4, f5, f6.
Does Hall’s condition hold for color class A?
4. At most how many faces can a 20-vertex planar graphGhave if every face of Gis bordered by at least 5 edges?
5. Which is the smallest positive integer whose number of divisors is divisible by 15?
6. Let us assume that the 500-vertex simple graph G has 450 vertices such that the degree of any of those vertices is at most 49. Prove that χ(G)≤50.
