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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 4colourable. 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.
 Spring '09
 M.PEI
 Math, Combinatorics

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