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# Graphing with Vertices

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6. Suppose G is a graph and &#61540;(G) &#61619; 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 &#61619; 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 &#61619; 3 ( &#61559;(G) &#61619; 3) or there is an independent set in G of size &#61619; 5 (&#61537;(G) &#61619; 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 &#61619; 5 or every vertex has degree &#61603; 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.

https://brainmass.com/math/graphs-and-functions/graphing-with-vertices-7013

#### 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 ...

#### Solution Summary

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