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Unformatted text preview: Planar Graphs II Margaret M. Fleck 30 April 2010 This lecture continues the discussion of planar graphs (section 9.7 of Rosen). 1 Announcements Makeup quiz last day of classes (at the start of class). Your room for the final exam (Friday the 7th, 710pm) is based on the first letter of your last name: AH Roger Adams Lab 116 IW Business Instructional Facility (515 E. Gregory opposite Armory) 1001 XZ Roger Adams Lab 116 The conflict exam (Monday the 10th, 1:304:30pm) is in 269 Everett Lab. 2 A corollary of Eulers formula Suppose G is a connected simple planar graph, with v vertices, e edges, and f faces, where v 3. Then e 3 v 6. 1 Proof: The sum of the degrees of the regions is equal to twice the number of edges. But each region must have degree 3. So we have 2 e 3 f . Then 2 3 e f . Eulers formula says that v e + f = 2, so f = e v +2. Combining this with 2 3 e f , we get e v + 2 2 3 e So e 3 v + 2 0. So e 3 v 2. Therefore e 3 v 6. We can also use this formula to show that the graph K 5 isnt planar. K 5 has five vertices and 10 edges. This isnt consistent with the formula e 3 v 6. Unfortunately, this trick doesnt work for K 3 , 3 , which isnt planar but satisfies the equation (with 6 vertices and 9 edges). 3 Another corollary In a similar way, we can show that if G is a connected planar simple graph with e edges and v vertices, with v 3, and if G has no circuits of length 3, then e 2 v 4. Proof: The sum of the degrees of the regions is equal to twice the number of edges. But each region must have degree 4 because we have no circuits of length 3. So we have 2 e 4 f . Then 1 2 e f ....
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This note was uploaded on 12/24/2010 for the course CS CS 173 taught by Professor Fleck during the Spring '10 term at University of Illinois, Urbana Champaign.
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