# ass6 - 3. (a) Prove that every planar graph without a...

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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 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.

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Unformatted text preview: 3. (a) Prove that every planar graph without a triangle (that is, a cycle of length 3) has a vertex of degree 3 or less. (b) Without using the four colour theorem, prove that every planar graph without a triangle is 4-colourable. 1 1 2 3 4 5 c d a b e 4. The Petersen graph is the graph depicted above, having ten vertices { 1 , 2 , 3 , 4 , 5 ,a,b,c,d,e } . (a) Show that the Petersen graph is not planar. (b) Show that the Petersen graph does not have any matching M and cover C with | M | = | C | . 5. Let M be a matching in a graph G , and let C be a cover of G . Suppose that every vertex in C is saturated by M . Prove that any augmenting path of M must have two consecutive vertices in C . 2...
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## This note was uploaded on 10/23/2009 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.

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ass6 - 3. (a) Prove that every planar graph without a...

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