Discrete Mathematics with Graph Theory (3rd Edition) 379

Discrete Mathematics with Graph Theory (3rd Edition) 379 -...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 13 (e) The graph is not planar, by Kuratowski's Theorem, since it contains K.3,3 as a subgraph, as shown. 3. (a) The graph is not planar. It has a subgraph homeomorphic to K. 3,3 as shown on the left. B G R 377 (b) Since the graph contains triangles, its chromatic number is at 3. A 3-coloring is shown on the right, so the chromatic number is 3. 4. (a) It cannot contain a subgraph homeomorphic to K.5 or K.3,3 because each of these graphs contain more than one circuit. (b) A near-tree is planar, so V - E + R = 2. There is just one circuit, so R = 2 and V = E. (c) We know that E deg v = 2/E/ = 2/V/. Suppose such a graph has a vertex of degree 3. If all others have degree greater than 2, then E deg v > 2/V/, a contradiction. 5. Exercise 6 of Section 13.1 tells us that 3E :::; 5V - 10. Substituting E = V + R - 2 gives the desired result. 6. (a) E :::;
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3V - 6 = 3(30) -6 = 84. (b) If the vertices of degree 1 are removed together with the edges with which they are incident, we are left with a subgraph 1i which has 15 vertices, and which is still connected and planar. The number of edges in 1i is at most 3(15) -6 = 39. Since 15 edges were removed from g, the number of edges in 9 is at most 39 + 15 = 54. 7. Here is K.2,3. This is homeomorphic to a graph with four vertices ("remove" a vertex of degree 2) and hence to a subgraph of K.4. 8. True. Since 9 and 1i are homeomorphic, each can be obtained from some graph K. by adding vertices of degree 2. If K. were not connected, then 9 would also not be connected. Also, if K. contained a circuit, so would g. Since 9 is a tree, it follows the K. must also be a tree. But introducing vertices of degree 2 to a tree still leaves a tree. So 1i is a tree....
View Full Document

This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

Ask a homework question - tutors are online