CIS 4930 Exam 2.
Name ______________________________
1. What is the chromatic number of the following graph:
2. Let G=(V, E) be a graph and f:V -> {1, 2} a function mapping the vertices of G to 1, 2
(that is, we label each vertex with either 1 or 2). Draw a connected graph with 5 vertices
and label each vertex with 1, 2 such that if u, v are adjacent, then f(u)=f(v).
3. Let G be a graph with chromatic number 2, that is, a bipartite graph. Argue that we can
orient the edges of G so that G has no directed path of length more than 1.
4. Let G be a graph with chromatic number 3, that is, a tri-partite graph. Argue that we
can orient the edges of G so that G has no directed path of length more than 2.
5. Let G be a graph with n vertices maximum independent set size k. Which of the
following are true?
a. The chromatic number of G is less than n/k
b. The chromatic number of G is greater than or equal to n/k
c. The chromatic number of G is less than or equal to n- k + 1.
d. The chromatic number of G is greater than n-k + 1
6. We know that the chromatic number of C5 is three and its maximum clique size is two.
What is the chromatic number and maximum clique size of the graph formed by taking
two copies of C5 and making each vertex in one adjacent to each vertex in the other?
7. Draw a planar triangulation with 10 vertices.
8. We know that K5 is not planar, nor is any complete graph with more than 5 vertices
planar. Let Kn-e denote the complete graph on n vertices minus any one edge. For which
values of n is this graph planar?
9. Draw an outerplanar triangulation with 8 vertices.
10. Let Ka, b, c be the complete tri-partite graph with the three parts having, a, b, and c
vertices (that is, any two vertices in different parts have an edge between them). Use
Kuratowski’s theorem to argue that K3, 2, 1 is not planar.