Graph Theory Practice Exam 2.
1. Draw a planar graph with 7 vertices that is a triangulation.
2. Let G be a 2-connected graph. Form graph G’ from G by adding a vertex v
that is adjacent to two vertices of G. Prove that G’ is 2-connected.
3. What is the chromatic number of the following graph?
4. How many different maximum length paths does C5 have?
5. Draw a planar embedding of K2,5
6. Let T be a tree with n vertices. Modify T by adding two edges. What are the
possible values of the chromatic number of T?
7. A homomorphism is a function f:V(G) -> V(H), that is, a function from the
vertices of graph G to the vertices of graph H, such that if uv is an edge in
G, then f(u)f(v) is an edge in H. Argue that if G has chromatic number k,
then there is a homomorphism from G to Kk.
8. Find a homomorphism from the following graph to C5.
9. What is the chromatic number of the m by n grid graph?
10. Explain and/or illustrate the concepts behind Brooks’ theorem, Erdos’
theorem, Kurastowski’s theorem, the four-color theorem.