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).
Solution: All vertices must have same label

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.
Orient all edges from A to B where A, B are the color classes
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.
Let A, B, C be color classes. Orient edges from A to B, B to C and A to C

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
T c.
The chromatic number of G is less than or equal to n- k + 1.
T d. The chromatic number of G is greater than n-k + 1
Since each color class must be an independent set, the number of colors needed is at least ceiling (n/k). For c, d, note that we can color all vertices in the maximum ind. set with one color and every other vertex its own color, using n-k+1 colors.
6. We know that the chromatic number of C
5
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 C
5
and making each vertex in one adjacent to each vertex in the other?
6 (this graph has 35 edges)
Clique size: 4

7. Draw a planar triangulation with 10 vertices.
8. We know that K
5
is not planar, nor is any complete graph with more than 5 vertices planar. Let K n
-e denote the complete graph on n vertices minus any one edge. For which values of n is this graph planar?
5, 4, 3, 2 (n=1 does not make sense)

9. Draw an outerplanar triangulation with 8 vertices.
This is a maximal outerplanar graph G such that each face is a triangle (should have 13 edges). I will accept if the outside face borders more than 3 edges (in which case graph may have less than 13 edges).
10. Let K a, 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 K
3, 2, 1 is not planar.
This graphs has a K_3, 3 subgraph which is not planar.