6.
Suppose G is a graph and (G)  n/3. Prove that G has one or two
connected components.
7.
a.
b.
c.
8.
Prove: given a graph G with 14 vertices, there is clique in G of size  3 (
(G)  3) or there is an independent set in G of size  5 ((G)  5).
Using notation from class, I’m asking you to give one half of the proof that
r (3,5) = 14.
You may use the fact that r(3,4) = 9.
Hint: Consider 2 cases- either there is vertex of degree  5 or every vertex
has degree  4.
9.
Suppose T is a tree with n vertices, an every vertex has degree 4 or is a
leaf. How many leaves does T have?
10.
Find the clique number an independence number of the following graph.
Prove your answers are correct.
Prove if n is odd, then there is no 3-regular graph with n vertices.
Give an example of a 3-regular graph with 8 vertices.
Prove: For every even n  4, there is a 3-regular graph with n
vertices.