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.
Vertex v is clearly not a cut-vertex of G’, since G is connected. Suppose G-u is not connected. Any two vertices w, x, neither equal to v or u, will have a path between them in G’ since G is 2-connected. Consider v, w, w not equal to u and let z be a neighbor of z not equal to u. There must exist a zw path in G-u, since G is 2connected. From, this path we c an easily make a wv path.
3.
What is the chromatic number of the following graph?
4.
How many different maximum length paths does C
5
have?
5
5.
Draw a planar embedding of K
2,5
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?
2, 3
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 K k
.
Done in class: each vertex of K k
corresponds to a color class.
8.
Find a homomorphism from the following graph to C
5.

9.
What is the chromatic number of the m by n grid graph?
2 (unless n=m=1)
10.
Explain and/or illustrate the concepts behind Brooks’ theorem, Erdos’ theorem, Kurastowski’s theorem, the four-color theorem.