Assignment 6 Soln

Assignment 6 Soln - MATH 239 Assignment 6 This assignment...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
MATH 239 Assignment 6 This assignment is due at noon on Friday, July 24, 2009, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. For each of the three connected graphs depicted below, determine if it is planar. If it is planar, exhibit a planar embedding. If it is not planar, exhibit a subgraph that is an edge subdivision of K 5 or K 3 , 3 . 2 1 3 4 5 6 A B C D E F G H t u v w x y z Solution: The first graph is not planar. It contains the following subgraph which is isomorphic to K 3 , 3 . 2 1 3 4 5 6 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The second graph is planar. A planar embedding is as follows. B D H F E G C A The third graph is not planar. It contains the following subgraph which is an edge subdivision of K 3 , 3 . t u v w x y z 2. Let G be a connected planar graph with p vertices, where p 3. Suppose that there exists a planar embedding of G having p faces. (a) Let q denote the number of edges in G . Show that q = 2 p - 2. (b) Prove that G is not 2-colourable. (c) Show that G is sometimes 3-colourable, by finding an example of such a graph G which is 3-colourable. Justify your answer. (d) Show that G is sometimes not 3-colourable, by finding an example of such a graph G which is not 3-colourable. Justify your answer. Solution: (a) From Euler’s formula, we have p - q + s = c + 1. Since G is connected, we have c = 1, and by hypothesis, we have s = p . Therefore p - q + p = 2, or q = 2 p - 2. (b) We first show that G is not a tree. A tree has 1 face in any planar embedding, and p 3 by hypothesis, so a tree cannot have p faces. Hence G is not a tree. We conclude that G has at least one cycle. Note that 2-colourable is equivalent to bipartite. If
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 08/09/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

Page1 / 5

Assignment 6 Soln - MATH 239 Assignment 6 This assignment...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online