By John M. Harris, Jeffry L. Hirst, Michael J. Mossinghoff (auth.)

ISBN-10: 1475748035

ISBN-13: 9781475748031

ISBN-10: 1475748051

ISBN-13: 9781475748055

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**Sample text**

Thus F = 6, and P is a cube. Case 3. Suppose r = 3 and k = 5. In this case we have 8 = V- F and 3 V = SF. Solving this system yields V = 20 and F = 12. This is a dodecahedron. Case 4. Suppose r = 4 and k = 3. Here we have F = 8 and 4V = 3F. Thus V = 6 and P is an octahedron. 42 1. Graph Theory Case 5. Suppose r = 5 and k = 3. In this case we have 8 = - V + F and 5V = 3F. Solving this system yields V = 12 and F = 20. This is an icosahedron. D Exercises 1. • , rn. 2. Find an example of a polyhedron different from the ones discussed in this section.

The sets S;. Case I. Suppose that G is not k-regular. Then there exists some vertex with degree less than k. Choose such a vertex and call it Vn. Let So = {Vn} and let 5 1 = N(vn), the neighborhood of Vn. 60). Since G is finite, there is some t such that 51 is not empty, and Sr is empty for all r > t. Next, label the vertices in S1 with the labels Vn-1· Vn-2· .. 1· Label the vertices in S2 with the labels Vn-151 1-1> ... 1-152 1· Continue labeling in this decreasing fashion until all vertices of G have been labeled.

B. c. d. e. R1 R2 R3 R4 Rs 3. Explain why embedding a graph in a plane is essentially the same as embedding a graph on a sphere. 4. Write a nice proof of the fact that every tree is planar. 5. Draw a planar graph in which every vertex has degree exactly 5. 6. Let G be a planar graph. Prove that G is bipartite if and only if b(R) is even for every region R. 7. In [50] and [145], Fary and Wagner proved independently that every planar graph has a planar representation in which every edge is a straight line segment.

### Combinatorics and Graph Theory by John M. Harris, Jeffry L. Hirst, Michael J. Mossinghoff (auth.)

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