Homework Three Quick Answers

- 1. Choose a max length path in G with endpoints a, b. Take a. Its links must go back into path, else path could be extended. Since it has d links, the link closest to b must be at least d away from a, creating a circuit of length d+1 or greater.
- 2. Consider (a) shortest path from a to b, where a, b not directly connected. If pathlength = 2: done. If longer, there can’t be any shortcuts, so no chord can be present.
- 3. Counterexample: K4.
- 4. Pictures.
- 5. Some vertex must have degree > N/2, which means that a bipartite split would divide the vertices into sets of size (N/2 + k), (N/2 -k), for k>0. This yields N**2/4 - k**2 edges, for k**2 > 0.
- 6. Pictures. Planar, start with two congruent triangles facing the same way, then add arcs to reach desired sequence. For nonplanar, start with K3,3, add two more edges.
- 7. 2N edges by handshaking lemma, so 10 = 2N - N +2, N = 8.
- 8. One of K, K-bar must have 28 edges, since K11 has 55 edges. This violates E<= 3V-6. If not connected then if planar, just add a connecting link between component(s).
- 9. Having a fourth circle, assuming general position, will violate R = E-V + 2.

`\-- Main.RobbieMoll - 2012-02-28`

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Topic revision: r2 - 2014-10-08 - RobbieMoll

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