2. Exercises ”Discrete Computational Biology”, WS 2011/12

Saarland University
Center for Bioinformatics
Dr. Marc Hellmuth
Á
2. Exercises ”Discrete Computational Biology”, WS 2011/12
Exercise 1: (5 Credits)
Give an example of an undirected graph G = (V,E) with |V| > 1 vertices such that:
(a) χ(G) = 1, (b) χ(G) = 2, (c) χ(G) = ∆(G)+1, (d) χ(G) = |V|.
Moreover, prove that χ(G) on your examples is correct.
Exercise 2: (5 Credits)
Let G = (V,E) be an undirected graph. Prove or disprove:
(a) If χ(G) = |V| then χ(G) = ∆(G)+1.
(b) If χ(G) = ∆(G)+1 then χ(G) = |V|.
Exercise 3: (5 Credits)
WritepseudocodeforanalgorithmthattestsifanundirectedgraphG = (V,E)isbipartite
and runs in O(|V|+|E|) time.
Exercise 4: (12+8=20 Credits)
Problem 3-COL: Given a graph G. Is G 3-colorable
Problem SCHED: Given a set of courses C that all take one time unit, a set of students
S, and a set R = {(S i ,C j ) | S i ∈ S wants to visit C j ∈ C}. Two courses C i and C j are
in conflict if there is some student S k ∈ S such that (S k ,C i ) ∈ R and (S k ,C j ) ∈ R. A
schedule is conflict-free if no two courses that are in conflict are scheduled at the same
time. Is there a conflict-free schedule requiring K time units
3-COL is NP-complete.
(a) Prove that SCHED is NP-complete.
(b) Let C = {C 1 ,...,C 5 } and S = {S 1 ,...,S 6 }. The courses the students plan to take
are:
student: S 1 S 2 S 3 S 4 S 5 S 6
courses: C 2 ,C 3 C 3 ,C 1 C 1 ,C 5 C 5 ,C 2 C 4 ,C 1 C 2 ,C 4
What is the minimal number of time units that are required for a conflict-free
schedule Prove your result.

Exercise 5: (5 Credits)
Let G = (V,E) be an undirected graph and G be its complement. Prove or disprove:
(a) If G is bipartite then G is bipartite.
(b) There are graphs with χ(G) = χ(G).
Deadline: Wednesday - Nov 16, 2011 - 2pm