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.