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Graphing with Vertices

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6. Suppose G is a graph and (G)  n/3. Prove that G has one or two connected components.

7. a. Prove if n is odd, then there is no 3-regular graph with n vertices.
b. Give an example of a 3-regular graph with 8 vertices.
c. Prove: For every even n  4, there is a 3-regular graph with n vertices.

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.

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Solution Preview

6.) for completeness of degree (N-1), no. of vertices: N

so for (n/3) degree, there should be at least (n/3 + 1) verices. Now let these (n/3 + 1) vertice make a sub graph.
Now, rest of the (2n/3 -1) vertices:
Again for (n/3) degree at least (n/3 + 1) vertices are needed. now left no. of vertices: (n/3 -2)
and definitely to be of degree n/3 degree these number of vertices ...

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