Practice Exam 1

Graph Theory Practice Exam.
1. A tree is a connected graph without any cycles. How many edges does a tree with
n vertices have?
2. A spanning tree of a graph G is a subgraph T of G that contains all the vertices of
G such that T is a tree. Prove, using induction on the number of vertices, that
every graph G contains a spanning tree. (Hint: in the inductive step, consider the
cases where the n+1st vertex, v, is or is not, a cut-vertex in the graph).
3. Draw K6. How many edges does its complement have?
4. Draw a 2-connected bipartite graph.
5. Draw a planar graph with 7 vertices that is a triangulation.
6. Let G be a 2-connected graph. Form graph G’ from G by adding a vertex v which
is adjacent to two vertices of G. Prove that G’ is 2-connected.
7. Draw a 3-regular graph having more than 4 vertices.
8. What is the chromatic number of the following graph?
9. Is the following graph bipartite?
10. How many maximum length paths does C5 have?