CS 594 Graph Theory, Spring 2014 Homework 4
1. What is the smallest graph that is not a line graph? Explain.
2. Prove or disprove:
a. If G has more than 2 vertices, and every pair of vertices belongs to a common cycle,
then G is 2-connected.
b. If G has more than 3 vertices, and every trio of vertices belongs to a common cycle,
then G is 3-connected.
3.
a. The Peterson graph is a 3-regular, 3-edge-connected graph in which every
disconnecting set creates an isolated vertex. Draw a 3-regular, 3-edge-connected graph with a
size 3 disconnecting set whose deletion does not create an isolated vertex.
b. Draw a 3-regular graph with edge-connectivity 2.
c. Draw a 3-regular graph with edge-connectivity 1.
d. What is the smallest order 2-edge-connected graph that contains a cut vertex?
4. Prove that if G is not a block, then G has at least two blocks.
5. Prove or disprove: If
is 2-connected, then
has at least two connected components.
6. Prove that if G has at least 2k vertices and is k-connected, then G has a cycle of length at least
2k.